reserve X for non empty set,
  Y for set,
  S for SigmaField of X,
  F for sequence of S,
  f,g for PartFunc of X,REAL,
  A,B for Element of S,
  r,s for Real,
  a for Real,
  n for Nat;

theorem Th20:
  R_EAL(r(#)f)=r(#)R_EAL f
proof
  dom(R_EAL(r(#)f)) = dom(R_EAL f) by VALUED_1:def 5;
  then
A1: dom(R_EAL(r(#)f)) = dom(r(#)R_EAL f) by MESFUNC1:def 6;
  now
    let x be object;
    reconsider rr= r as R_eal by XXREAL_0:def 1;
    assume
A2: x in dom R_EAL(r(#)f);
    then (R_EAL(r(#)f)).x = r*(f.x) by VALUED_1:def 5;
    then (R_EAL(r(#)f)).x = rr * f.x by EXTREAL1:1;
    hence (R_EAL(r(#)f)).x = (r(#)R_EAL f).x by A1,A2,MESFUNC1:def 6;
  end;
  hence thesis by A1,FUNCT_1:2;
end;
