
theorem Th1:
  for x,y,z being set st z in x & z in y holds x \ {z} = y \ {z} implies x = y
proof
  let x,y,z be set;
  assume that
A1: z in x and
A2: z in y;
  assume
A3: x \ {z} = y \ {z};
  thus x = x \/ {z} by A1,ZFMISC_1:40
    .= (y \ {z}) \/ {z} by A3,XBOOLE_1:39
    .= y \/ {z} by XBOOLE_1:39
    .= y by A2,ZFMISC_1:40;
end;
