reserve X,Y,Z,x,y,z for set;
reserve T,R for Tolerance of X;

theorem
  for Y being set for T,R being Tolerance of union Y st
  (for x,y being object holds [x,y] in T iff
     ex Z st Z in Y & x in Z & y in Z) &
  (for x,y being object holds [x,y] in R iff
  ex Z st Z in Y & x in Z & y in Z) holds T = R
proof
  let Y be set;
  let T,R be Tolerance of union Y such that
A1: for x,y being object holds [x,y] in T iff
   ex Z st Z in Y & x in Z & y in Z and
A2: for x,y being object holds [x,y] in R iff
   ex Z st Z in Y & x in Z & y in Z;
  for x,y being object holds [x,y] in T iff [x,y] in R
  proof
    let x,y be object;
    thus [x,y] in T implies [x,y] in R
    proof
      assume [x,y] in T;
      then ex Z st Z in Y & x in Z & y in Z by A1;
      hence thesis by A2;
    end;
    assume [x,y] in R;
    then ex Z st Z in Y & x in Z & y in Z by A2;
    hence thesis by A1;
  end;
  hence thesis;
end;
