reserve a,b,r for Real;
reserve A,B for non empty set;
reserve f,g,h for Element of PFuncs(A,REAL);

theorem Th18:
  (multrealpfunc A).(a,(multrealpfunc A).(b,f)) = (multrealpfunc A ).(a*b,f)
proof
  reconsider aa=a, bb=b as Element of REAL by XREAL_0:def 1;
  reconsider g = (multrealpfunc A).(bb,f) as Element of PFuncs(A,REAL);
  reconsider h = (multrealpfunc A).(aa,g) as Element of PFuncs(A,REAL);
  reconsider k = (multrealpfunc A).(aa*bb,f) as Element of PFuncs(A,REAL);
A1: dom h = dom g by Th9;
A2: dom g = dom f by Th9;
A3: now
    let x be Element of A;
    assume
A4: x in dom h;
    hence h.x =a*(g.x) by A1,Th9
      .=a*(b*(f.x)) by A2,A1,A4,Th9
      .=(a*b)*(f.x)
      .= k.x by A2,A1,A4,Th9;
  end;
  dom k = dom f by Th9;
  hence thesis by A2,A1,A3,PARTFUN1:5;
end;
