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 Th37:
  X1 in Solutions_of(A,B1) & X2 in Solutions_of(A,B2) & width B1 =
  width B2 implies X1 + X2 in Solutions_of(A,B1 + B2)
proof
  assume that
A1: X1 in Solutions_of(A,B1) and
A2: X2 in Solutions_of(A,B2) and
A3: width B1 = width B2;
A4: ex Y1 be Matrix of K st Y1 = X1 & len Y1 = width A & width Y1 = width B1
  & A * Y1 = B1 by A1;
A5: width (X1+X2) = width X1 by MATRIX_3:def 3;
A6: len (X1+X2) = len X1 by MATRIX_3:def 3;
A7: ex Y2 be Matrix of K st Y2 = X2 & len Y2 = width A & width Y2 = width B2
  & A * Y2 = B2 by A2;
A8: A*(X1+X2)=A*X1 + A*X2 by A3,A4,A7,MATRIX_4:62;
  width (B1+B2) = width B1 by MATRIX_3:def 3;
  hence thesis by A4,A7,A6,A5,A8;
end;
