
theorem Th35:
for X be non empty set, f be PartFunc of X,ExtREAL, r1,r2 be Real
 holds r1(#)(r2(#)f) = (r1*r2)(#)f
proof
   let X be non empty set, f be PartFunc of X,ExtREAL, r1,r2 be Real;
A1:dom(r1(#)(r2(#)f)) = dom(r2(#)f) by MESFUNC1:def 6; then
A2:dom(r1(#)(r2(#)f)) = dom f by MESFUNC1:def 6;
A3:dom((r1*r2)(#)f) = dom f by MESFUNC1:def 6;
   for x be Element of X st x in dom(r1(#)(r2(#)f)) holds
    (r1(#)(r2(#)f)).x = ((r1*r2)(#)f).x
   proof
    let x be Element of X;
    assume A4: x in dom(r1(#)(r2(#)f)); then
    (r1(#)(r2(#)f)).x = r1 * (r2(#)f).x by MESFUNC1:def 6; then
A5: (r1(#)(r2(#)f)).x = r1 * (r2 * f.x) by A1,A4,MESFUNC1:def 6;
    ((r1*r2)(#)f).x = (r1*r2)*f.x by A2,A3,A4,MESFUNC1:def 6;
    hence thesis by A5,XXREAL_3:66;
   end;
   hence r1(#)(r2(#)f) = (r1*r2)(#)f by A2,A3,PARTFUN1:5;
end;
