theorem Th24:
  cell(x,x) = {x}
proof
  for x9 being object holds x9 in cell(x,x) iff x9 = x
  proof
    let x9 be object;
    thus x9 in cell(x,x) implies x9 = x
    proof
      assume
A1:   x9 in cell(x,x);
      then reconsider x,x9 as Function of Seg d,REAL by Def3;
      now
        let i;
A2:     for i holds x.i <= x.i;
        then
A3:     x.i <= x9.i by A1,Th21;
        x9.i <= x.i by A1,A2,Th21;
        hence x9.i = x.i by A3,XXREAL_0:1;
      end;
      hence thesis by FUNCT_2:63;
    end;
    thus thesis by Th23;
  end;
  hence thesis by TARSKI:def 1;
end;
