reserve x,y,z,a,b,c,X,A for set;
reserve C,D for Coherence_Space;
reserve T for Tolerance of union C;

theorem
  (for x,y st x in a & y in a holds {x,y} in C) implies a c= union C
proof
  assume
A1: for x,y st x in a & y in a holds {x,y} in C;
  let x be object;
  assume x in a;
  then {x,x} in C by A1;
  then
A2: {x} in C by ENUMSET1:29;
  x in {x} by TARSKI:def 1;
  hence thesis by A2,TARSKI:def 4;
end;
