reserve G, G1, G2 for _Graph, H for Subgraph of G;

theorem Th31:
  for F being PGraphMapping of G1, G2
  st F is weak_SG-embedding holds SG2SGFunc(F) is one-to-one
proof
  let F be PGraphMapping of G1, G2;
  assume A1: F is weak_SG-embedding;
  now
    let x1,x2 be object;
    set f = SG2SGFunc(F);
    assume A2: x1 in dom f & x2 in dom f & f.x1 = f.x2;
    then reconsider H1 = x1, H2 = x2 as plain Subgraph of G1 by Th1;
    A3: rng(F | H1) = f.x1 by Def5
      .= rng(F | H2) by A2, Def5;
    A4: F | H1 is total & F | H2 is total by A1, GLIB_010:57;
    A5: F_V.:the_Vertices_of H1
       = rng(F_V | the_Vertices_of H1) by RELAT_1:115
      .= the_Vertices_of rng(F | H1) by A4, GLIB_010:54
      .= rng(F|H2)_V by A3, A4, GLIB_010:54
      .= F_V.:the_Vertices_of H2 by RELAT_1:115;
    the_Vertices_of H1 c= the_Vertices_of G1 &
      the_Vertices_of H2 c= the_Vertices_of G1;
    then the_Vertices_of H1 c= dom F_V & the_Vertices_of H2 c= dom F_V
      by A1, GLIB_010:def 11;
    then A6: the_Vertices_of H1 c= the_Vertices_of H2 &
      the_Vertices_of H2 c= the_Vertices_of H1 by A1, A5, FUNCT_1:87;
    A7: F_E.:the_Edges_of H1
       = rng(F_E | the_Edges_of H1) by RELAT_1:115
      .= the_Edges_of rng(F | H1) by A4, GLIB_010:54
      .= rng(F|H2)_E by A3, A4, GLIB_010:54
      .= F_E.:the_Edges_of H2 by RELAT_1:115;
    the_Edges_of H1 c= the_Edges_of G1 &
      the_Edges_of H2 c= the_Edges_of G1;
    then the_Edges_of H1 c= dom F_E & the_Edges_of H2 c= dom F_E
      by A1, GLIB_010:def 11;
    then the_Edges_of H1 c= the_Edges_of H2 &
      the_Edges_of H2 c= the_Edges_of H1 by A1, A7, FUNCT_1:87;
    then H1 is Subgraph of H2 & H2 is Subgraph of H1 by A6, GLIB_000:44;
    hence x1 = x2 by GLIB_000:87, GLIB_009:44;
  end;
  hence thesis by FUNCT_1:def 4;
end;
