
theorem Th4:
  for X being set,
      A being Subset of X,
      F being A-subsetpreserving BinOp of X holds
    F||A is BinOp of A
proof
  let X be set;
  let A be Subset of X;
  let F be A-subsetpreserving BinOp of X;
  dom F = [:X,X:] by PARTFUN1:def 2; then
A1: dom (F||A) = [:A,A:] by RELAT_1:62,ZFMISC_1:96;
  for x being object holds x in [:A,A:] implies F||A.x in A
  proof
    let x be object;
    assume
A2: x in [:A,A:];
    then F||A.x=F.x by A1,FUNCT_1:47;
    hence thesis by A2,Def4;
  end;
  hence thesis by A1,FUNCT_2:3;
end;
