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

theorem
  F is_even_on A implies for x st x in A holds F.x = F. |.x.|
proof
  assume
A1: F is_even_on A;
  then
A2: A c= dom F;
A3: F|A is even by A1;
  let x such that
A4: x in A;
A5: x in dom(F|A) by A2,A4,RELAT_1:62;
A6: -x in A by A4,Def1;
  then
A7: -x in dom(F|A) by A2,RELAT_1:62;
      reconsider x as Element of REAL by XREAL_0:def 1;
  per cases;
  suppose
    x < 0;
    then F. |.x.|=F.(-x) by ABSVALUE:def 1
      .=F/.(-x) by A2,A6,PARTFUN1:def 6
      .=F|A/.(-x) by A2,A6,PARTFUN2:17
      .=F|A.(-x) by A7,PARTFUN1:def 6
      .=F|A.x by A3,A5,A7,Def3
      .=F|A/.x by A5,PARTFUN1:def 6
      .=F/.x by A2,A4,PARTFUN2:17
      .=F.x by A2,A4,PARTFUN1:def 6;
    hence thesis;
  end;
  suppose
    0 < x;
    hence thesis by ABSVALUE:def 1;
  end;
  suppose
    x = 0;
    hence thesis by ABSVALUE:def 1;
  end;
end;
