
theorem Th26:
  for R,S being Abelian add-associative right_zeroed
  right_complementable associative distributive well-unital non empty
doubleLoopStr, F being non empty Subset of R, P being Function of R,S st P is
  RingIsomorphism holds P.:(F-Ideal) = (P.:F)-Ideal
proof
  let R,S being Abelian add-associative right_zeroed right_complementable
  associative distributive well-unital non empty doubleLoopStr, F being non
  empty Subset of R, P being Function of R,S;
  assume
A1: P is RingIsomorphism;
  now
    let x be object;
A2: dom P = the carrier of R by FUNCT_2:def 1;
    assume
A3: x in (P.:F)-Ideal;
    then consider lc being LinearCombination of P.: F such that
A4: x = Sum lc by IDEAL_1:60;
    consider E be FinSequence of [:the carrier of S, the carrier of S, the
    carrier of S:] such that
A5: E represents lc by IDEAL_1:35;
    P is RingMonomorphism by A1;
    then
A6: P is one-to-one;
A7: P is onto by A1;
    then
A8: (P qua Function)" = P" by A6,TOPS_2:def 4;
    (P qua Function)" .: (P.:F) = (P qua Function)"(P.:F) by A6,FUNCT_1:85;
    then consider lc9 being LinearCombination of F such that
A9: len lc = len lc9 & for i being set st i in dom lc9 holds lc9.i =
    (P".((E/.i)`1_3))*(P".((E /.i)`2_3))*(P".((E/.i)`3_3))
      by A6,A8,A5,FUNCT_1:82,IDEAL_1:36;
    P" is RingIsomorphism by A1,Th25;
    then P" is RingMonomorphism;
    then P" is RingHomomorphism;
    then P".x = Sum lc9 by A4,A5,A9,Th24;
    then P".x in F-Ideal by IDEAL_1:60;
    then P.(P".x) in P.:(F-Ideal) by A2,FUNCT_1:def 6;
    hence x in P.:(F-Ideal) by A3,A6,A7,A8,FUNCT_1:35;
  end;
  then
A10: (P.:F)-Ideal c= P.:(F-Ideal);
  now
    let x be object;
    assume x in P.:(F-Ideal);
    then consider x9 being object such that
    x9 in the carrier of R and
A11: x9 in F-Ideal and
A12: x = P.x9 by FUNCT_2:64;
    consider lc9 being LinearCombination of F such that
A13: x9 = Sum lc9 by A11,IDEAL_1:60;
    consider E being FinSequence of [:the carrier of R, the carrier of R, the
    carrier of R:] such that
A14: E represents lc9 by IDEAL_1:35;
    consider lc being LinearCombination of P.:F such that
A15: len lc9 = len lc & for i being set st i in dom lc
    holds lc.i = (P.((E/.i)`1_3))* (P.((E/.i)`2_3))*(P.((E/.i)`3_3))
        by A14,IDEAL_1:36;
    P is RingMonomorphism by A1;
    then P is RingHomomorphism;
    then x = Sum lc by A12,A13,A14,A15,Th24;
   hence x in (P.:F)-Ideal by IDEAL_1:60;
  end;
  then P.:(F-Ideal) c= (P.:F)-Ideal;
  hence thesis by A10,XBOOLE_0:def 10;
end;
