reserve x,y for set;
reserve D for non empty set;
reserve UN for Universe;

theorem Th1:
  for G1,G2,G3 being non empty doubleLoopStr, f being Function of
  G1,G2, g being Function of G2,G3 st f is linear & g is linear holds g*f is
  linear
proof
  let G1,G2,G3 be non empty doubleLoopStr, f be Function of G1,G2, g be
  Function of G2,G3 such that
A1: f is additive multiplicative unity-preserving and
A2: g is additive multiplicative unity-preserving;
  set h = g*f;
  thus h is additive
  proof
    let x,y be Scalar of G1;
A3: g.(f.x) = h.x & g.(f.y) = h.y by FUNCT_2:15;
    thus h.(x+y) = g.(f.(x+y)) by FUNCT_2:15
      .= g.(f.x+f.y) by A1
      .= h.x+h.y by A2,A3;
  end;
  thus h is multiplicative
  proof
    let x,y be Scalar of G1;
A4: g.(f.x) = h.x & g.(f.y) = h.y by FUNCT_2:15;
    thus h.(x*y) = g.(f.(x*y)) by FUNCT_2:15
      .= g.(f.x*f.y) by A1
      .= h.x*h.y by A2,A4;
  end;
  thus h.(1_G1) = g.(f.(1_G1)) by FUNCT_2:15
    .= g.(1_G2) by A1
    .= 1_G3 by A2;
end;
