reserve X for set,
        n,m,k for Nat,
        K for Field,
        f for n-element real-valued FinSequence,
        M for Matrix of n,m,F_Real;

theorem Th21:
  for F be one-to-one FinSequence of TOP-REAL n st
    rng F is linearly-independent
  for B be OrdBasis of n-VectSp_over F_Real st B = MX2FinS 1.(F_Real,n)
  for M be Matrix of n,F_Real st M is invertible & M|len F = F
  holds (Mx2Tran M).:[#]Lin rng(B|len F) = [#]Lin rng F
proof
  let F be one-to-one FinSequence of TOP-REAL n such that
   A1: rng F is linearly-independent;
  reconsider f=F as FinSequence of n-VectSp_over F_Real by Lm1;
  set MF=FinS2MX f;
  set n1=n-' len F;
  set L=len F;
  lines MF is linearly-independent by A1,Th7;
  then the_rank_of MF=len F by MATRIX13:121;
  then L<=width MF by MATRIX13:74;
  then A2: L<=n by MATRIX_0:23;
  then A3: n-L=n1 by XREAL_1:233;
  set V=n-VectSp_over F_Real;
  let B be OrdBasis of n-VectSp_over F_Real such that
   A4: B=MX2FinS 1.(F_Real,n);
  let M be Matrix of n,F_Real such that
   M is invertible and
   A5: M|len F=F;
  consider q being FinSequence such that
   A6: M=F^q by A5,FINSEQ_1:80;
  M=MX2FinS M;
  then reconsider q as FinSequence of n-VectSp_over F_Real by A6,FINSEQ_1:36;
  A7: len M=len F+len q by A6,FINSEQ_1:22;
  set Mq=FinS2MX q;
  set MT=Mx2Tran M;
  A8: len M=n by MATRIX_0:def 2;
  A9: dom MT=[#]TOP-REAL n by FUNCT_2:52;
   A10: dom Mx2Tran MF=[#]TOP-REAL L by FUNCT_2:def 1;
   A11: the carrier of TOP-REAL n=REAL n by EUCLID:22
    .=n-tuples_on REAL;
   A12: rng(Mx2Tran MF)=[#]Lin lines MF by Th18
    .=[#]Lin rng F by Th6;
   A13: (n|->0)=0*n
    .=0.TOP-REAL n by EUCLID:70;
   A14: (n1|->0)=0*n1
    .=0.TOP-REAL n1 by EUCLID:70;
   then A15: (Mx2Tran Mq).(n1|->0)=0.TOP-REAL n by A3,A8,A7,MATRTOP1:29;
   thus MT.:[#]Lin rng(B|L)c=[#]Lin rng F
   proof
    let y be object;
    assume y in MT.:[#]Lin rng(B|L);
    then consider x be object such that
     A16: x in dom MT and
     A17: x in [#]Lin rng(B|L) and
     A18: MT.x=y by FUNCT_1:def 6;
    reconsider x as Element of TOP-REAL n by A16;
    len x=n by CARD_1:def 7;
    then len(x|L)=L by A2,FINSEQ_1:59;
    then A19: x|L is L-element by CARD_1:def 7;
    then A20: x|L is Element of TOP-REAL L by Lm3;
    A21: (Mx2Tran MF).(x|L) is Element of n-tuples_on REAL by A11,A19,Lm3;
    x in Lin rng(B|L) by A17;
    then x=(x|L)^((n-' L) |->0) by A4,Th20;
    then y=(Mx2Tran MF).(x|L)+(Mx2Tran Mq).(n1|->0)
      by A3,A8,A6,A7,A18,A19,MATRTOP1:36
     .=(Mx2Tran MF).(x|L) by A13,A15,A21,RVSUM_1:16;
    hence thesis by A12,A10,A20,FUNCT_1:def 3;
   end;
   let y be object;
   assume y in [#]Lin rng F;
   then consider x be object such that
    A22: x in dom(Mx2Tran MF) and
    A23: (Mx2Tran MF).x=y by A12,FUNCT_1:def 3;
   reconsider x as Element of TOP-REAL L by A22;
   (Mx2Tran MF).x is Element of TOP-REAL n by Lm3;
   then A24: y=(Mx2Tran MF).x+0.TOP-REAL n by A11,A13,A23,RVSUM_1:16
    .=(Mx2Tran MF).x+(Mx2Tran Mq).(n1|->0) by A3,A8,A7,A14,MATRTOP1:29
    .=MT.(x^(n1|->0)) by A6,A3,A8,A7,MATRTOP1:36;
   set xx=(x^(n1|->0));
   len x=L by CARD_1:def 7;
   then dom x=Seg L by FINSEQ_1:def 3;
   then xx=(xx|L)^(n1|->0) by FINSEQ_1:21;
   then xx in Lin rng(B|L) by A4,A3,Th20;
   then A25: xx in [#]Lin rng(B|L);
   xx is Element of TOP-REAL n by A3,Lm3;
   hence thesis by A9,A24,A25,FUNCT_1:def 6;
end;
