reserve x,y for set,
  i,j,k,l,m,n for Nat,
  K for Field,
  N for without_zero finite Subset of NAT,
  a,b for Element of K,
  A,B,B1,B2,X,X1,X2 for (Matrix of K),
  A9 for (Matrix of m,n,K),
  B9 for (Matrix of m,k,K);
reserve D for non empty set,
  bD for FinSequence of D,
  b,f,g for FinSequence of K,
  MD for Matrix of D;

theorem
  for A be Matrix of n,m,K, B be Matrix of n,k,K for P be Function of
Seg n,Seg n holds Solutions_of(A,B) c= Solutions_of(A*P,B*P) & (P is one-to-one
  implies Solutions_of(A,B) = Solutions_of(A*P,B*P))
proof
  set IDn=idseq n;
  len IDn=n & IDn is FinSequence of NAT by CARD_1:def 7,FINSEQ_2:48;
  then reconsider IDn as Element of n-tuples_on NAT by FINSEQ_2:92;
  let A be Matrix of n,m,K, B be Matrix of n,k,K;
  let P be Function of Seg n,Seg n;
A1: rng P c= Seg n by RELAT_1:def 19;
  dom IDn=Seg n;
  then reconsider IDnP=IDn*P as FinSequence of NAT by FINSEQ_2:47;
  dom P=Seg n by FUNCT_2:52;
  then n in NAT & dom IDnP = Seg n by A1,ORDINAL1:def 12,RELAT_1:53;
  then len IDnP = n by FINSEQ_1:def 3;
  then reconsider IDnP as Element of n-tuples_on NAT by FINSEQ_2:92;
A2: n=len A by MATRIX_0:def 2;
A3: (idseq n)*P = P by A1,RELAT_1:53;
  then
A4: rng IDnP c=dom A by A1,A2,FINSEQ_1:def 3;
A5: IDn=Sgm Seg n & card Seg n=n by FINSEQ_1:57,FINSEQ_3:48;
  then
A6: Segm(A,IDnP,Sgm Seg width A) = Segm(A,Seg len A,Seg width A) * P by A2,
MATRIX13:33
    .= A*P by MATRIX13:46;
A7: len B=n by MATRIX_0:def 2;
  then
A8: Segm(B,IDnP,Sgm Seg width B) = Segm(B,Seg len B,Seg width B) * P by A5,
MATRIX13:33
    .= B*P by MATRIX13:46;
  per cases;
  suppose
A9: n>0;
    hence Solutions_of(A,B)c=Solutions_of(A*P,B*P) by A6,A8,A4,Th42;
A10: card Seg n = card Seg n;
A11: dom A=Seg n by A2,FINSEQ_1:def 3;
A12: dom B=Seg n by A7,FINSEQ_1:def 3;
    assume P is one-to-one;
    then P is onto by A10,FINSEQ_4:63;
    then rng P=Seg n by FUNCT_2:def 3;
    then for i st i in (dom A) \ rng IDnP holds Line(A,i) = width A |-> 0.K &
    Line(B,i) = width B |-> 0.K by A3,A11,XBOOLE_1:37;
    hence thesis by A1,A3,A6,A8,A9,A11,A12,Th43;
  end;
  suppose
A13: n=0;
    then len B=0 by MATRIX_0:22;
    then
A14: B={};
    len A=0 by A13,MATRIX_0:22;
    then
A15: A={};
    A*P={} by A13;
    hence thesis by A15,A14;
  end;
end;
