theorem Th53:
  for M1,M2 being Matrix of D st M1@=M2@ & len M1=len M2 holds M1 = M2
proof
  let M1,M2 be Matrix of D;
  assume that
A1: M1@=M2@ and
A2: len M1=len M2;
  len (M1@) = width M1 by Def6;
  then
A3: width M1=width M2 by A1,Def6;
A4: Indices M2=[:dom M2,Seg width M2:];
  for i,j st [i,j] in Indices M1 holds M1*(i,j) = M2*(i,j)
  proof
    let i,j;
    assume
A5: [i,j] in Indices M1;
    dom M1 = Seg len M2 by A2,FINSEQ_1:def 3
      .= dom M2 by FINSEQ_1:def 3;
    then M2@*(j,i) = M2*(i,j) by A3,A4,A5,Def6;
    hence thesis by A1,A5,Def6;
  end;
  hence thesis by A2,A3,Th21;
end;
