
theorem Th5:
  for A being non trivial set
  for x being Element of A
  for F being (A \ {x})-subsetpreserving BinOp of A holds
    F||(A\{x}) is BinOp of A \ {x}
proof
  let A be non trivial set;
  let x be Element of A;
  let F be (A \ {x})-subsetpreserving BinOp of A;
  dom F = [:A,A:] by FUNCT_2:def 1; then
A1: dom(F||(A\{x})) = [:A\{x},A\{x}:] by RELAT_1:62,ZFMISC_1:96;
  for y being object holds y in [:A\{x},A\{x}:] implies F||(A\{x}).y in A\{x}
  proof
    let y be object;
    assume
A2: y in [:A\{x},A\{x}:];
    then F||(A\{x}).y=F.y by A1,FUNCT_1:47;
    hence thesis by A2,Def4;
  end;
  hence thesis by A1,FUNCT_2:3;
end;
