 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 Th34:
  Mx2Tran M1 = Mx2Tran M2 implies M1 = M2
proof
  assume that
   A1: Mx2Tran M1=Mx2Tran M2;
  set Vn=n-VectSp_over F_Real,Vm=m-VectSp_over F_Real;
  reconsider Bn=MX2FinS 1.(F_Real,n) as OrdBasis of Vn by MATRLIN2:45;
  reconsider Bm=MX2FinS 1.(F_Real,m) as OrdBasis of Vm by MATRLIN2:45;
A2: len Bm=m by Th19;
  len Bn=n by Th19;
  then reconsider A1=M1,B1=M2 as Matrix of len Bn,len Bm,F_Real by A2;
  A3: Mx2Tran(A1,Bn,Bm)=Mx2Tran M1 by Th20
   .=Mx2Tran(B1,Bn,Bm) by A1,Th20;
  thus M1=AutMt(Mx2Tran(A1,Bn,Bm),Bn,Bm) by MATRLIN2:36
   .=M2 by A3,MATRLIN2:36;
end;
