 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
  M = 1.(F_Real,m) |n implies ((Mx2Tran M).f) |n = f
proof
  set ONE=1.(F_Real,m);
  assume A1: M=ONE|n;
  per cases;
  suppose A2: n=0;
   then ((Mx2Tran M).f) |n is empty;
   hence thesis by A2;
  end;
  suppose A3: n>0;
   set TRm=TOP-REAL m;
   set V=m-VectSp_over F_Real;
   A4: len ONE=m by MATRIX_0:24;
   A5: len f=n by CARD_1:def 7;
   consider A be FinSequence such that
    A6: ONE=M^A by A1,FINSEQ_1:80;
   ONE=MX2FinS ONE;
   then reconsider A as FinSequence of V by A6,FINSEQ_1:36;
   set L=len A;
   len M=n by A3,MATRIX13:1;
   then n+L=m by A4,A6,FINSEQ_1:22;
   then A7: f^(L|->0) is Element of TRm by Lm2;
   set A1=FinS2MX A;
   A8: (f^(L|->0)) |n=(f^(L|->0)) |dom f by A5,FINSEQ_1:def 3
    .=f by FINSEQ_1:21;
   (Mx2Tran(M^A1)).(f^(L|->0))=(Mx2Tran ONE).(f^(L|->0))
      by A3,A4,A6,MATRIX13:1
    .=(id TRm).(f^(L|->0)) by Th33
    .=f^(L|->0) by A7;
   hence thesis by A8,Th35;
  end;
end;
