
theorem Th21:
  for X being set st 4 c= card X for w,x,y being object ex z being object
  st z in X & w<>z & x<>z & y<>z
proof
  let X be set;
  assume 4 c= card X;
  then consider a,b,c,d being object such that
A1: a in X and
A2: b in X and
A3: c in X and
A4: d in X and
A5: a<>b & a<>c & a<>d & b<>c & b<>d & d<>c by Th20;
  let w,x,y be object;
  per cases;
  suppose
    w <> a & x <> a & y <> a;
    hence thesis by A1;
  end;
  suppose
    w <> a & x <> a & y = a & w <> b & x <> b;
    hence thesis by A2,A5;
  end;
  suppose
    w <> a & x <> a & y = a & w <> b & x = b & w <> c;
    hence thesis by A3,A5;
  end;
  suppose
    w <> a & x <> a & y = a & w <> b & x = b & w = c;
    hence thesis by A4,A5;
  end;
  suppose
    w <> a & x <> a & y = a & w = b & x <> c;
    hence thesis by A3,A5;
  end;
  suppose
    w <> a & x <> a & y = a & w = b & x = c;
    hence thesis by A4,A5;
  end;
  suppose
    w <> a & x = a & w <> b & y <> b;
    hence thesis by A2,A5;
  end;
  suppose
    w <> a & x = a & w <> b & y = b & w <> c;
    hence thesis by A3,A5;
  end;
  suppose
    w <> a & x = a & w <> b & y = b & w = c;
    hence thesis by A4,A5;
  end;
  suppose
    w <> a & x = a & w = b & y <> c;
    hence thesis by A3,A5;
  end;
  suppose
    w <> a & x = a & w = b & y = c;
    hence thesis by A4,A5;
  end;
  suppose
    w = a & x <> b & y <> b;
    hence thesis by A2,A5;
  end;
  suppose
    w = a & x <> b & y = b & x <> c;
    hence thesis by A3,A5;
  end;
  suppose
    w = a & x <> b & y = b & x = c;
    hence thesis by A4,A5;
  end;
  suppose
    w = a & x = b & y <> c;
    hence thesis by A3,A5;
  end;
  suppose
    w = a & x = b & y = c;
    hence thesis by A4,A5;
  end;
end;
