 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 Th33:
  Mx2Tran 1.(F_Real,n) = id TOP-REAL n
proof
  set V=n-VectSp_over F_Real;
  reconsider Bn=MX2FinS 1.(F_Real,n) as OrdBasis of V by MATRLIN2:45;
  A1: len Bn=n by Th19;
  then reconsider ONE=1.(F_Real,n) as Matrix of len Bn,len Bn,F_Real;
  A2: Mx2Tran 1.(F_Real,n)=Mx2Tran(ONE,Bn,Bn) by Th20;
  A3: [#]TOP-REAL n=dom Mx2Tran 1.(F_Real,n) by FUNCT_2:def 1
   .=[#]V by A2,FUNCT_2:def 1;
  thus Mx2Tran 1.(F_Real,n)
   =Mx2Tran(AutMt(id V,Bn,Bn),Bn,Bn) by A1,Th20,MATRLIN2:28
   .=id TOP-REAL n by A3,MATRLIN2:34;
end;
