
theorem
  for G being _Graph, C being Component of G holds
    G.minDegree() c= C.minDegree() &
    G.minInDegree() c= C.minInDegree() &
    G.minOutDegree() c= C.minOutDegree()
proof
  let G be _Graph, C being Component of G;
  A1: the_Vertices_of C c= the_Vertices_of G;
  thus G.minDegree() c= C.minDegree()
  proof
    set A = the set of all v.degree() where v is Vertex of G;
    set B = the set of all w.degree() where w is Vertex of C;
    A2: B <> {}
    proof
      set w = the Vertex of C;
      w.degree() in B;
      hence thesis;
    end;
    now
      let x be object;
      assume x in B;
      then consider w being Vertex of C such that
        A3: x = w.degree();
      reconsider v = w as Vertex of G by A1, TARSKI:def 3;
      x = v.degree() by A3, GLIBPRE0:44;
      hence x in A;
    end;
    then meet A c= meet B by A2, SETFAM_1:6, TARSKI:def 3;
    then G.minDegree() c= meet B;
    hence thesis;
  end;
  thus G.minInDegree() c= C.minInDegree()
  proof
    set A = the set of all v.inDegree() where v is Vertex of G;
    set B = the set of all w.inDegree() where w is Vertex of C;
    A4: B <> {}
    proof
      set w = the Vertex of C;
      w.inDegree() in B;
      hence thesis;
    end;
    now
      let x be object;
      assume x in B;
      then consider w being Vertex of C such that
        A5: x = w.inDegree();
      reconsider v = w as Vertex of G by A1, TARSKI:def 3;
      x = v.inDegree() by A5, GLIBPRE0:44;
      hence x in A;
    end;
    then meet A c= meet B by A4, SETFAM_1:6, TARSKI:def 3;
    then G.minInDegree() c= meet B;
    hence thesis;
  end;
  thus G.minOutDegree() c= C.minOutDegree()
  proof
    set A = the set of all v.outDegree() where v is Vertex of G;
    set B = the set of all w.outDegree() where w is Vertex of C;
    A6: B <> {}
    proof
      set w = the Vertex of C;
      w.outDegree() in B;
      hence thesis;
    end;
    now
      let x be object;
      assume x in B;
      then consider w being Vertex of C such that
        A7: x = w.outDegree();
      reconsider v = w as Vertex of G by A1, TARSKI:def 3;
      x = v.outDegree() by A7, GLIBPRE0:44;
      hence x in A;
    end;
    then meet A c= meet B by A6, SETFAM_1:6, TARSKI:def 3;
    then G.minOutDegree() c= meet B;
    hence thesis;
  end;
end;
