reserve x, r for Real;
reserve A for symmetrical Subset of COMPLEX;
reserve F,G for PartFunc of REAL, REAL;

theorem Th20:
  F is_odd_on A implies |. F .| is_even_on A
proof
  assume
A1: F is_odd_on A;
  then
A2: A c= dom F;
  then
A3: A c= dom (|. F .|) by VALUED_1:def 11;
  then
A4: dom((|. F .|)|A) = A by RELAT_1:62;
A5: F|A is odd by A1;
  for x st x in dom((|. F .|)|A) & -x in dom((|. F .|)|A) holds (|. F .|)|
  A.(-x)=(|. F .|)|A.x
  proof
    let x;
    assume that
A6: x in dom((|. F .|)|A) and
A7: -x in dom((|. F .|)|A);
A8: x in dom(F|A) by A2,A4,A6,RELAT_1:62;
A9: -x in dom(F|A) by A2,A4,A7,RELAT_1:62;
      reconsider x as Element of REAL by XREAL_0:def 1;
    (|. F .|)|A.(-x)=(|. F .|)|A/.(-x) by A7,PARTFUN1:def 6
      .=(|. F .|)/.(-x) by A3,A4,A7,PARTFUN2:17
      .=(|. F .|).(-x) by A3,A7,PARTFUN1:def 6
      .=|. F.(-x) .| by A3,A7,VALUED_1:def 11
      .=|. F/.(-x) .| by A2,A7,PARTFUN1:def 6
      .=|. F|A/.(-x) .| by A2,A4,A7,PARTFUN2:17
      .=|. F|A.(-x) .| by A9,PARTFUN1:def 6
      .=|. -F|A.x .| by A5,A8,A9,Def6
      .=|. -F|A/.x .| by A8,PARTFUN1:def 6
      .=|. -F/.x .| by A2,A4,A6,PARTFUN2:17
      .=|. -F.x .| by A2,A6,PARTFUN1:def 6
      .=|. F.x .| by COMPLEX1:52
      .=(|. F .|).x by A3,A6,VALUED_1:def 11
      .=(|. F .|)/.x by A3,A6,PARTFUN1:def 6
      .=(|. F .|)|A/.x by A3,A4,A6,PARTFUN2:17
      .=(|. F .|)|A.x by A6,PARTFUN1:def 6;
    hence thesis;
  end;
  then (|. F .|)|A is with_symmetrical_domain quasi_even by A4;
  hence thesis by A3;
end;
