
theorem
  for G1,G2 being _finite connected real-weighted WGraph, G3 being
  WSubgraph of G1 st G3 is minimumSpanningTree of G1 & G1 == G2 & the_Weight_of
  G1 = the_Weight_of G2 holds G3 is minimumSpanningTree of G2
proof
  let G1,G2 be _finite connected real-weighted WGraph, G3 be WSubgraph of G1;
  assume that
A1: G3 is minimumSpanningTree of G1 and
A2: G1 == G2 and
A3: the_Weight_of G1 = the_Weight_of G2;
  set G39 = G3;
  reconsider G39 as Tree-like WSubgraph of G2 by A1,A2,A3,GLIB_003:10;
  the_Vertices_of G3 = the_Vertices_of G1 by A1,GLIB_000:def 33
    .= the_Vertices_of G2 by A2;
  then reconsider G39 as spanning Tree-like WSubgraph of G2 by GLIB_000:def 33;
  now
    let G be spanning Tree-like WSubgraph of G2;
    reconsider G9=G as Tree-like WSubgraph of G1 by A2,A3,GLIB_003:10;
    the_Vertices_of G = the_Vertices_of G2 by GLIB_000:def 33
      .= the_Vertices_of G1 by A2;
    then G9 is spanning;
    hence G3.cost() <= G.cost() by A1,Def19;
  end;
  then G39 is min-cost;
  hence thesis;
end;
