How to get independent increment of equation numbers

Question

I want the increment of the equation counter to be independent of other increments.

The setup I have :

Theorem 1.1

Equation (1.2)

Corollary 1.3

Equation (1.4)

Lemma 1.5

Lemma 1.6

Proposition 1.7

Equation (1.8)

What I really want :

Theorem 1.1

Equation (1.1)

Corollary 1.2

Equation (1.2)

Lemma 1.3

Lemma 1.4

Proposition 1.5

Equation (1.3)

In the preamble I have :

\documentclass[11pt,a4paper,reqno]{amsart}
\usepackage{amssymb}
\usepackage{bm}
\numberwithin{equation}{section}
\usepackage{mathabx}
\usepackage{mathtools}

\theoremstyle{plain}
\newtheorem{theorem}[equation]{Theorem}
\newtheorem{proposition}[equation]{Proposition}
\newtheorem{lemma}[equation]{Lemma}
\newtheorem{corollary}[equation]{Corollary}
\newtheorem{conjecture}[equation]{Conjecture}
\newtheorem{principle}[equation]{Principle}
\newtheorem{claim}[equation]{Claim}

\theoremstyle{definition}
\newtheorem{definition}[equation]{Definition}
\newtheorem{remark}[equation]{Remark}
\newtheorem{example}[equation]{Example}

Any help would be much appreciated.

Answer 1

First, change

\newtheorem{theorem}[equation]{Theorem}

to

\newtheorem{theorem}{Theorem}
\numberwithin{theorem}{section}

Or, more succinctly,

\newtheorem{theorem}{Theorem}[section] % Thanks, @Bernard! 

Second, change

\newtheorem{proposition}[equation]{Proposition}
\newtheorem{lemma}[equation]{Lemma}
\newtheorem{corollary}[equation]{Corollary}
\newtheorem{conjecture}[equation]{Conjecture}
\newtheorem{principle}[equation]{Principle}
\newtheorem{claim}[equation]{Claim}
\theoremstyle{definition}
\newtheorem{definition}[equation]{Definition}
\newtheorem{remark}[equation]{Remark}
\newtheorem{example}[equation]{Example}

to

\newtheorem{proposition}[theorem]{Proposition}
\newtheorem{lemma}[theorem]{Lemma}
\newtheorem{corollary}[theorem]{Corollary}
\newtheorem{conjecture}[theorem]{Conjecture}
\newtheorem{principle}[theorem]{Principle}
\newtheorem{claim}[theorem]{Claim}
\theoremstyle{definition}
\newtheorem{definition}[theorem]{Definition}
\newtheorem{remark}[theorem]{Remark}
\newtheorem{example}[theorem]{Example}