reserve c,c1,c2 for Cardinal, G,G1,G2 for _Graph, v for Vertex of G;

theorem
  for G being c-regular _Graph holds 2*`G.size() = c*`G.order()
proof
  let G be c-regular _Graph;
  per cases;
  suppose c is empty;
    then A1: c*`G.order() = 0 & G is edgeless by CARD_2:20;
    then G.size() = 0;
    hence thesis by A1, CARD_2:20;
  end;
  suppose A2: c is non empty;
    A3: for A being set st A in P(G) holds card A = c
    proof
      let A be set;
      assume A in P(G);
      then consider v being Vertex of G such that
        A4: A = E0(G,v) \/ E1(G,v) & v is non isolated;
      thus thesis by A4, Lm11;
    end;
    thus 2*`G.size() = G.size()*`card {0,1} by CARD_2:57
      .= card [: card the_Edges_of G, card {0,1} :] by CARD_2:def 2
      .= card [: the_Edges_of G, {0,1} :] by CARD_2:7
      .= card dom F(G) by Lm9
      .= card rng F(G) by Lm8, CARD_1:70
      .= card union P(G) by Lm16
      .= c*`card P(G) by A3, Lm14, Th56
      .= c*`G.order() by A2, Lm15;
  end;
end;
