let a, b be Real; :: thesis: for A being non empty set
for f being Element of PFuncs (A,REAL) holds (multrealpfunc A) . (a,((multrealpfunc A) . (b,f))) = (multrealpfunc A) . ((a * b),f)
let A be non empty set ; :: thesis: for f being Element of PFuncs (A,REAL) holds (multrealpfunc A) . (a,((multrealpfunc A) . (b,f))) = (multrealpfunc A) . ((a * b),f)
let f be Element of PFuncs (A,REAL); :: thesis: (multrealpfunc A) . (a,((multrealpfunc A) . (b,f))) = (multrealpfunc A) . ((a * b),f)
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 :: thesis: for x being Element of A st x in dom h holds
h . x = k . x

let x be Element of A; :: thesis: ( x in dom h implies h . x = k . x )
assume A4: x in dom h ; :: thesis: h . x = k . x
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 ;
:: thesis: verum

end;

dom k = dom f by Th9;
hence (multrealpfunc A) . (a,((multrealpfunc A) . (b,f))) = (multrealpfunc A) . ((a * b),f) by A2, A1, A3, PARTFUN1:5; :: thesis: verum