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 Th80:
  f|Y is constant implies f|Y is bounded
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;
  now
    take p=|.r.|;
    let c;
    assume c in Y /\ dom f;
    hence |.(f/.c).| <= p by A1;
  end;
  hence thesis by Th68;
end;
