
theorem Th4:
for x1,y1,x2,y2 being object, X being set
 st x1 in X & x2 in X & {x1,[y1,X]} = {x2,[y2,X]}
  holds x1 = x2 & y1 = y2
proof
 let x1,y1,x2,y2 be object,X be set;
 assume that A1: x1 in X and  A2: x2 in X;
 assume A3: {x1,[y1,X]} = {x2,[y2,X]};
 per cases by A3,ZFMISC_1:6;
 suppose x1 = x2 & [y1,X] = [y2,X];
  hence x1 = x2 & y1 = y2 by XTUPLE_0:1;
 end;
 suppose x1 = x2 & [y1,X] = x2;
  hence x1 = x2 & y1 = y2 by Th1,A2;
 end;
 suppose x1 = [y2,X] & [y1,X] = x2;
  hence x1 = x2 & y1 = y2 by Th1,A2;
 end;
 suppose x1 = [y2,X] & [y1,X] = [y2,X];
  hence x1 = x2 & y1 = y2 by Th1,A1;
 end;
end;
