
theorem Th9:
  for X be non empty set, f,g be PartFunc of X,ExtREAL,
      a be ExtReal, r be Real
    st r <> 0 & g = r(#)f holds eq_dom(f,a) = eq_dom(g,a*r)
proof
    let X be non empty set, f,g be PartFunc of X,ExtREAL,
    a be ExtReal, r be Real;
    assume that
A1:  r <> 0 and
a2:  g = r(#)f;
A2: dom f = dom g by a2,MESFUNC1:def 6;
    now let x be object;
     assume x in eq_dom(f,a); then
     x in dom f & f.x = a by MESFUNC1:def 15; then
     x in dom g & g.x = r*a by a2,A2,MESFUNC1:def 6;
     hence x in eq_dom(g,a*r) by MESFUNC1:def 15;
    end; then
A3: eq_dom(f,a) c= eq_dom(g,a*r);
    now let x be object;
     assume x in eq_dom(g,a*r); then
A4:  x in dom g & g.x = a*r by MESFUNC1:def 15; then
     r* f.x = a*r by a2,MESFUNC1:def 6; then
     f.x = a by A1,XXREAL_3:68;
     hence x in eq_dom(f,a) by A2,A4,MESFUNC1:def 15;
    end; then
    eq_dom(g,a*r) c= eq_dom(f,a);
    hence thesis by A3;
end;
