reserve x,y,X,Y for set;
reserve C for non empty set;
reserve c for Element of C;
reserve f,f1,f2,f3,g,g1 for PartFunc of C,COMPLEX;
reserve r1,r2,p1 for Real;
reserve r,q,cr1,cr2 for Complex;

theorem Th79:
  f|Y is constant implies |.f.||Y is constant & (-f)|Y is constant
proof
  assume f|Y is constant;
  then consider r being Element of COMPLEX such that
A1: for c st c in Y /\ dom f holds (f/.c) = r by PARTFUN2:35;
  reconsider rr= |.r.| as Element of REAL by XREAL_0:def 1;
  now
    let c;
    assume
A2: c in Y /\ dom (|.f.|);
    then c in dom (|.f.|) by XBOOLE_0:def 4;
    then
A3: c in dom f by VALUED_1:def 11;
    c in Y by A2,XBOOLE_0:def 4;
    then
A4: c in Y /\ dom f by A3,XBOOLE_0:def 4;
    f.c = f/.c by A3,PARTFUN1:def 6;
    hence (|.f.|).c = |.(f/.c).| by VALUED_1:18
      .= rr by A1,A4;
  end;
  hence |.f.||Y is constant by PARTFUN2:57;
A5: -r in COMPLEX by XCMPLX_0:def 2;
  now
    let c;
    assume
A6: c in Y /\ dom (-f);
    then c in Y /\ dom f by Th5;
    then
A7: -(f/.c) = -r by A1;
    c in dom (-f) by A6,XBOOLE_0:def 4;
    hence (-f)/.c = -r by A7,Th5;
  end;
  hence thesis by A5,PARTFUN2:35;
end;
