 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 Th20:
  for Bn be OrdBasis of n-VectSp_over F_Real,
      Bm be OrdBasis of m-VectSp_over F_Real st
    Bn = MX2FinS 1.(F_Real,n) & Bm = MX2FinS 1.(F_Real,m)
  for M1 be Matrix of len Bn,len Bm,F_Real st M1 = M
    holds Mx2Tran M = Mx2Tran(M1,Bn,Bm)
proof
  let Bn be OrdBasis of n-VectSp_over F_Real,
      Bm be OrdBasis of m-VectSp_over F_Real such that
   A1: Bn=MX2FinS 1.(F_Real,n) and
   A2: Bm=MX2FinS 1.(F_Real,m);
  set T=Mx2Tran M;
  let M1 be Matrix of len Bn,len Bm,F_Real such that
   A3: M1=M;
  set Tb=Mx2Tran(M1,Bn,Bm);
  dom Tb=the carrier of n-VectSp_over F_Real by FUNCT_2:def 1;
  then A4: dom Tb =REAL n by MATRIX13:102
   .=the carrier of TOP-REAL n by EUCLID:22;
  A5: for x be object st x in dom T holds T.x=Tb.x
  proof
   let x be object;
   assume x in dom T;
   then reconsider v=x as Element of TOP-REAL n by FUNCT_2:def 1;
   reconsider g=v as Vector of n-VectSp_over F_Real by A4,FUNCT_2:def 1;
   set L=LineVec2Mx(@v);
   len v=n by CARD_1:def 7;
   then A6: len L=1 & width L=n by MATRIX13:1;
   A7: len Bn=n by A1,Th19;
   A8: len Bm=m by A2,Th19;
   per cases;
   suppose A9: n=0;
    then Tb.g = 0.(m-VectSp_over F_Real) by A1,Th19,MATRLIN2:33
     .= m |-> 0.F_Real by MATRIX13:102
     .= 0* m
     .= 0.TOP-REAL m by EUCLID:70
     .= T.v by A9,Def3;
    hence thesis;
   end;
   suppose A10: n>0;
    A11: (Tb.g) |--Bm=Tb.g by A2,A8,MATRLIN2:46;
    A12: g|--Bn=g & AutMt(Tb,Bn,Bm)=M by A1,A3,A7,MATRLIN2:36,46;
    1 in dom L & len M=width L by A10,A6,FINSEQ_3:25,MATRIX13:1;
    then LineVec2Mx(Line(L*M,1))=LineVec2Mx(Line(L,1))*M by MATRLIN2:35
     .=L*M by MATRIX15:25
     .=LineVec2Mx(Tb.g|--Bm) by A7,A10,A12,MATRLIN2:31;
    hence Tb.x=Line(LineVec2Mx(Line(L*M,1)),1) by A11,MATRIX15:25
     .=Line(L*M,1) by MATRIX15:25
     .=T.x by A10,Def3;
   end;
  end;
  dom T=the carrier of TOP-REAL n by FUNCT_2:def 1;
  hence thesis by A4,A5,FUNCT_1:2;
end;
