reserve x,x1,x2,x3,x4,x5,x6,x7,x8,x9,x10,y for object, X,Z for set;

theorem :: from AMI_7, 2006.03.15, A.T.
  for x, y, z being set st x <> y & x <> z holds {x, y, z} \ {x} = {y, z }
proof
  let x, y, z be set such that
A1: x <> y & x <> z;
  hereby
    let a be object;
    assume
A2: a in {x, y, z} \ {x};
    then a in {x, y, z} by XBOOLE_0:def 5;
    then
A3: a = x or a = y or a = z by Def1;
    not a in {x} by A2,XBOOLE_0:def 5;
    hence a in {y, z} by A3,TARSKI:def 1,def 2;
  end;
  let a be object;
  assume a in {y, z};
  then
A4: a = y or a = z by TARSKI:def 2;
  then
A5: not a in {x} by A1,TARSKI:def 1;
  a in {x, y, z} by A4,Def1;
  hence thesis by A5,XBOOLE_0:def 5;
end;
