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

theorem Th81:
  B c= dom (sec) implies sec is_even_on B
proof
  assume
A1: B c= dom (sec);
  then
A2: dom(sec|B) = B by RELAT_1:62;
A3: for x st x in B holds sec.(-x) = sec.x
  proof
    let x;
    assume
A4: x in B;
    then -x in B by Def1;
    then sec.(-x)=(cos.(-x))" by A1,RFUNCT_1:def 2
      .=(cos.x)" by SIN_COS:30
      .= sec.x by A1,A4,RFUNCT_1:def 2;
    hence thesis;
  end;
  for x st x in dom(sec|B) & -x in dom(sec|B) holds sec|B.(-x)=sec|B.x
  proof
    let x;
    assume that
A5: x in dom(sec|B) and
A6: -x in dom(sec|B);
    sec|B.(-x)=sec|B/.(-x) by A6,PARTFUN1:def 6
      .=sec/.(-x) by A1,A2,A6,PARTFUN2:17
      .=sec.(-x) by A1,A6,PARTFUN1:def 6
      .=sec.x by A3,A5
      .=sec/.x by A1,A5,PARTFUN1:def 6
      .=sec|B/.x by A1,A2,A5,PARTFUN2:17
      .=sec|B.x by A5,PARTFUN1:def 6;
    hence thesis;
  end;
  then sec|B is with_symmetrical_domain quasi_even by A2;
  hence thesis by A1;
end;
