reserve x1,x2,z for set;
reserve A,B for non empty set;
reserve f,g,h for Element of Funcs(A,REAL);
reserve a,b for Real;

theorem
  (RealFuncExtMult A).(a,f) = a (#) f
proof
   reconsider aa=a as Element of REAL by XREAL_0:def 1;
   set h = (RealFuncExtMult A).(a,f);
b1:dom (a (#) f) = dom f by VALUED_1:def 5 .= A by FUNCT_2:def 1;
A2:now let x be Element of A;
     thus h.x = (multreal[;](aa,f)).x by Def3
      .= multreal.(aa,f.x) by FUNCOP_1:53
      .= a*(f.x) by BINOP_2:def 11
      .= (a (#) f).x by VALUED_1:def 5,b1;
   end;
   a in REAL & f in Funcs(A,REAL) by XREAL_0:def 1; then
   [a,f] in [:REAL,Funcs(A,REAL):] by ZFMISC_1:87; then
   (RealFuncExtMult A).(a,f) is Function of A, REAL by FUNCT_2:66,5; then
B1:dom ((RealFuncExtMult A).(a,f)) = A by FUNCT_2:def 1;
   for x being object st x in dom (a (#) f) holds
     ((RealFuncExtMult A).(a,f)).x = (a (#) f).x by A2,b1;
   hence thesis by FUNCT_1:2,B1,b1;
end;
