reserve E,V for set, G,G1,G2 for _Graph, c,c1,c2 for Cardinal, n for Nat;
reserve f for VColoring of G;
reserve g for EColoring of G;

theorem Th87:
  for g9 being one-to-one Function, g2 being EColoring of G
  st g2 = g9*g & g is proper holds g2 is proper
proof
  let g9 be one-to-one Function, g2 be EColoring of G;
  assume A1: g2 = g9*g & g is proper;
  now
    let v be Vertex of G;
    set V = v.edgesInOut();
    A2: g|V is one-to-one by A1;
    g2|V = g9*(g|V) by A1, RELAT_1:83;
    hence g2|V is one-to-one by A2;
  end;
  hence thesis;
end;
