
theorem Th2:
  for X be non empty set, A be set, r be Real, f be PartFunc of X,ExtREAL
    holds (r(#)f)|A = r(#)(f|A)
proof
  let X be non empty set, A be set, r be Real, f be PartFunc of X,ExtREAL;
A1: dom(r(#)f) = dom f by MESFUNC1:def 6; then
A2: dom((r(#)f)|A) = dom f /\ A by RELAT_1:61; then
A3: dom((r(#)f)|A) = dom(f|A) by RELAT_1:61; then
A4: dom((r(#)f)|A) = dom(r(#)(f|A)) by MESFUNC1:def 6;
    now let x be Element of X;
     assume B1: x in dom((r(#)f)|A); then
B2:  x in dom f by A2,XBOOLE_0:def 4;
     ((r(#)f)|A).x = (r(#)f).x by B1,FUNCT_1:47; then
     ((r(#)f)|A).x = r * f.x by B2,A1,MESFUNC1:def 6; then
     ((r(#)f)|A).x = r * (f|A).x by B1,A3,FUNCT_1:47;
     hence ((r(#)f)|A).x = (r(#)(f|A)).x by B1,A4,MESFUNC1:def 6;
    end;
    hence (r(#)f)|A = r(#)(f|A) by A4,PARTFUN1:5;
end;
