reserve GS for GraphStruct;
reserve G,G1,G2,G3 for _Graph;
reserve e,x,x1,x2,y,y1,y2,E,V,X,Y for set;
reserve n,n1,n2 for Nat;
reserve v,v1,v2 for Vertex of G;

theorem Th114:
  for G1 being _Graph, v being set, G2 being removeVertex of G1, v
  st G1 is _trivial or not v in the_Vertices_of G1 holds G1 == G2
proof
  let G1 be _Graph, v be set, G2 be removeVertex of G1, v;
  assume G1 is _trivial or not v in the_Vertices_of G1;
  then per cases;
  suppose G1 is _trivial;
    then consider v1 being Vertex of G1 such that
      A1: the_Vertices_of G1 = {v1} by Th22;
    set V = the_Vertices_of G1 \ {v}, E = G1.edgesBetween(V);
    per cases;
    suppose V is non empty Subset of the_Vertices_of G1 &
        E c= G1.edgesBetween(V);
      then V = {v1} by A1, ZFMISC_1:33;
      hence thesis by A1, Lm10, ZFMISC_1:57;
    end;
    suppose not (V is non empty Subset of the_Vertices_of G1 &
        E c= G1.edgesBetween(V));
      hence thesis by Def37;
    end;
  end;
  suppose not v in the_Vertices_of G1;
    hence thesis by Lm10;
  end;
end;
