
theorem Th41:
  for G being _finite connected real-weighted WGraph, G1 being
  minimumSpanningTree of G, G2 being WGraph st G1 == G2 & the_Weight_of G1 =
  the_Weight_of G2 holds G2 is minimumSpanningTree of G
proof
  let G be _finite connected real-weighted WGraph, G1 be minimumSpanningTree of
  G, G2 be WGraph;
  assume that
A1: G1 == G2 and
A2: the_Weight_of G1 = the_Weight_of G2;
  reconsider G29=G2 as WSubgraph of G by A1,A2,GLIB_003:8;
  the_Vertices_of G2 = the_Vertices_of G1 by A1
    .= the_Vertices_of G by GLIB_000:def 33;
  then reconsider G29 as spanning Tree-like WSubgraph of G by A1,
GLIB_000:def 33,GLIB_002:48;
  now
    let G3 be spanning Tree-like WSubgraph of G;
    G1.cost() <= G3.cost() by Def19;
    hence G29.cost() <= G3.cost() by A1,A2;
  end;
  hence thesis by Def19;
end;
