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
  a <> 0.K implies Solutions_of(A,B) = Solutions_of(a*A,a*B)
proof
  assume
A1: a<>0.K;
  thus Solutions_of(A,B) c= Solutions_of(a*A,a*B)
  proof
    let x be object such that
A2: x in Solutions_of(A,B);
    ex X st x=X & len X = width A & width X = width B & A * X = B by A2;
    hence thesis by A2,Th35;
  end;
A3: a"*(a*A) = (a"*a)*A by Th2
    .= 1_K *A by A1,VECTSP_1:def 10
    .= A by Th2;
A4: a"*(a*B) = (a"*a)*B by Th2
    .= 1_K*B by A1,VECTSP_1:def 10
    .= B by Th2;
  let x be object such that
A5: x in Solutions_of(a*A,a*B);
  ex X st x=X & len X = width (a*A) & width X = width (a* B) & (a*A) * X =
  (a*B) by A5;
  hence thesis by A5,A3,A4,Th35;
end;
