
theorem Th62:
  for G1, G2 being _Graph st G1 == G2 holds
    G1.supDegree() = G2.supDegree() &
    G1.minDegree() = G2.minDegree() &
    G1.supInDegree() = G2.supInDegree() &
    G1.minInDegree() = G2.minInDegree() &
    G1.supOutDegree() = G2.supOutDegree() &
    G1.minOutDegree() = G2.minOutDegree()
proof
  let G1, G2 be _Graph;
  assume A1: G1 == G2;
  set S1 = the set of all v.degree() where v is Vertex of G1;
  set S2 = the set of all v.degree() where v is Vertex of G2;
  now
    let x be object;
    hereby
      assume x in S1;
      then consider v1 being Vertex of G1 such that
        A2: x = v1.degree();
      reconsider v2 = v1 as Vertex of G2 by A1, GLIB_000:def 34;
      x = v2.degree() by A1, A2, GLIB_000:96;
      hence x in S2;
    end;
    assume x in S2;
    then consider v2 being Vertex of G2 such that
      A3: x = v2.degree();
    reconsider v1 = v2 as Vertex of G1 by A1, GLIB_000:def 34;
    x = v1.degree() by A1, A3, GLIB_000:96;
    hence x in S1;
  end;
  hence G1.supDegree() = G2.supDegree() &
    G1.minDegree() = G2.minDegree() by TARSKI:2;
  set S3 = the set of all v.inDegree() where v is Vertex of G1;
  set S4 = the set of all v.inDegree() where v is Vertex of G2;
  now
    let x be object;
    hereby
      assume x in S3;
      then consider v1 being Vertex of G1 such that
        A4: x = v1.inDegree();
      reconsider v2 = v1 as Vertex of G2 by A1, GLIB_000:def 34;
      x = v2.inDegree() by A1, A4, GLIB_000:96;
      hence x in S4;
    end;
    assume x in S4;
    then consider v2 being Vertex of G2 such that
      A5: x = v2.inDegree();
    reconsider v1 = v2 as Vertex of G1 by A1, GLIB_000:def 34;
    x = v1.inDegree() by A1, A5, GLIB_000:96;
    hence x in S3;
  end;
  hence G1.supInDegree() = G2.supInDegree() &
    G1.minInDegree() = G2.minInDegree() by TARSKI:2;
  set S5 = the set of all v.outDegree() where v is Vertex of G1;
  set S6 = the set of all v.outDegree() where v is Vertex of G2;
  now
    let x be object;
    hereby
      assume x in S5;
      then consider v1 being Vertex of G1 such that
        A6: x = v1.outDegree();
      reconsider v2 = v1 as Vertex of G2 by A1, GLIB_000:def 34;
      x = v2.outDegree() by A1, A6, GLIB_000:96;
      hence x in S6;
    end;
    assume x in S6;
    then consider v2 being Vertex of G2 such that
      A7: x = v2.outDegree();
    reconsider v1 = v2 as Vertex of G1 by A1, GLIB_000:def 34;
    x = v1.outDegree() by A1, A7, GLIB_000:96;
    hence x in S5;
  end;
  hence G1.supOutDegree() = G2.supOutDegree() &
    G1.minOutDegree() = G2.minOutDegree() by TARSKI:2;
end;
