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 Th35:
  X in Solutions_of(A,B) implies a*X in Solutions_of(A,a*B) & X in
  Solutions_of(a*A,a*B)
proof
A1: width (a*B) = width B & width (a*A)=width A by MATRIX_3:def 5;
  assume X in Solutions_of(A,B);
  then consider X1 such that
A2: X = X1 & len X1 = width A and
A3: width X1 = width B & A * X1 = B;
A4: len (a*X) = width A & width (a*X) = width X1 by A2,MATRIX_3:def 5;
  A*(a*X)=a*(A*X) & (a*A)*X=a*(A*X) by A2,Th1,MATRIXR1:22;
  hence thesis by A2,A3,A4,A1;
end;
