 reserve X,Y for set,
         n,m,k,i for Nat,
         r for Real,
         R for Element of F_Real,
         K for Field,
         f,f1,f2,g1,g2 for FinSequence,
         rf,rf1,rf2 for real-valued FinSequence,
         cf,cf1,cf2 for complex-valued FinSequence,
         F for Function;
reserve f,f1,f2 for n-element real-valued FinSequence,
        p,p1,p2 for Point of TOP-REAL n,
        M,M1,M2 for Matrix of n,m,F_Real,
        A,B for Matrix of n,F_Real;

theorem Th43:
  for A,B st Det A <> 0.F_Real holds (Mx2Tran A)" = Mx2Tran B iff A~ = B
proof
  A1: n=0 implies n=0;
  let A,B such that
   A2: Det A<>0.F_Real;
  A3: A is invertible by A2,LAPLACE:34;
  set MA=Mx2Tran A,MB=Mx2Tran B;
  A4: width B=n by MATRIX_0:24;
  reconsider ma=MA,mb=MB as Function;
  A5: width A=n & len A=n by MATRIX_0:24;
  A6: MA is one-to-one by A2,Th40;
  hereby assume MA"=MB;
   then A7: mb*ma=id dom ma by A6,FUNCT_1:39
    .=id TOP-REAL n by FUNCT_2:def 1
    .=Mx2Tran 1.(F_Real,n) by Th33;
   MB*MA=Mx2Tran(A*B) by A5,A4,A1,Th32;
   then B is_reverse_of A by A3,A7,Th34,MATRIX_6:38;
   hence A~=B by A3,MATRIX_6:def 4;
  end;
  assume A8: A~=B;
  MA is onto by A2,Th42;
  then A9: dom MB=[#]TOP-REAL n & rng MA=[#]TOP-REAL n by FUNCT_2:def 1,def 3;
  A10: n in NAT by ORDINAL1:def 12;
  MB*MA=Mx2Tran(A*B) by A5,A4,A1,Th32
   .=Mx2Tran 1.(F_Real,n) by A3,A10,A8,MATRIX14:18
   .=id TOP-REAL n by Th33
   .=id dom MA by FUNCT_2:def 1;
  hence thesis by A6,A9,FUNCT_1:41;
end;
