reserve A,B,a,b,c,d,e,f,g,h for set;

theorem
  id {a,b,c,d} = {[a,a],[b,b],[c,c],[d,d]}
proof
  set X = {a,b,c,d};
  thus id X c= {[a,a],[b,b],[c,c],[d,d]}
  proof
    let x be object;
    assume
A1: x in id X;
    then consider x1,y1 being object such that
A2: x = [x1,y1] and
A3: x1 in X and
    y1 in X by RELSET_1:2;
A4: x1 = y1 by A1,A2,RELAT_1:def 10;
    per cases by A3,ENUMSET1:def 2;
    suppose
      x1 = a;
      hence thesis by A2,A4,ENUMSET1:def 2;
    end;
    suppose
      x1 = b;
      hence thesis by A2,A4,ENUMSET1:def 2;
    end;
    suppose
      x1 = c;
      hence thesis by A2,A4,ENUMSET1:def 2;
    end;
    suppose
      x1 = d;
      hence thesis by A2,A4,ENUMSET1:def 2;
    end;
  end;
  let x be object;
  assume
A5: x in {[a,a],[b,b],[c,c],[d,d]};
  per cases by A5,ENUMSET1:def 2;
  suppose
A6: x = [a,a];
    a in X by ENUMSET1:def 2;
    hence thesis by A6,RELAT_1:def 10;
  end;
  suppose
A7: x = [b,b];
    b in X by ENUMSET1:def 2;
    hence thesis by A7,RELAT_1:def 10;
  end;
  suppose
A8: x = [c,c];
    c in X by ENUMSET1:def 2;
    hence thesis by A8,RELAT_1:def 10;
  end;
  suppose
A9: x = [d,d];
    d in X by ENUMSET1:def 2;
    hence thesis by A9,RELAT_1:def 10;
  end;
end;
