
theorem
  for G1, G2, G3 being WGraph, F1 being PGraphMapping of G1, G2,
    F2 being PGraphMapping of G2, G3
  st F1 is weight-preserving & F2 is weight-preserving
  holds F2 * F1 is weight-preserving
proof
  let G1, G2, G3 be WGraph;
  let F1 be PGraphMapping of G1, G2, F2 be PGraphMapping of G2, G3;
  assume A1: F1 is weight-preserving & F2 is weight-preserving;
  thus the_Weight_of G3 * (F2 * F1)_E
     = (the_Weight_of G3 * F2_E) * F1_E by RELAT_1:36
    .= (the_Weight_of G1) | dom (F2 * F1)_E  by A1, Th1;
end;
