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 Th78:
  f|Y is constant implies (q(#)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;
A2: q*r in COMPLEX by XCMPLX_0:def 2;
  now
    let c;
    assume
A3: c in Y /\ dom (q(#)f);
    then
A4: c in Y by XBOOLE_0:def 4;
A5: c in dom (q(#)f) by A3,XBOOLE_0:def 4;
    then c in dom f by Th4;
    then
A6: c in Y /\ dom f by A4,XBOOLE_0:def 4;
    thus (q(#)f)/.c = q * (f/.c) by A5,Th4
      .= q*r by A1,A6;
  end;
  hence thesis by A2,PARTFUN2:35;
end;
