reserve i,n,m for Nat;

theorem Th6:
for f be PartFunc of REAL m,REAL n,
    g be PartFunc of REAL-NS m,REAL-NS n, a be Real st
 f = g holds a(#)f = a(#)g
proof
   let f be PartFunc of REAL m,REAL n,
       g be PartFunc of REAL-NS m,REAL-NS n,
       a be Real;
   assume A1: f = g;
   the carrier of REAL-NS m = REAL m
   & the carrier of REAL-NS n = REAL n by REAL_NS1:def 4; then
   reconsider aG = a(#)g as PartFunc of REAL m,REAL n;
A2:dom f = dom aG by A1,VFUNCT_1:def 4;
A3: a(#)f = f[#]a by INTEGR15:def 11;
   for c be object st c in dom aG holds aG .c = a (#) (f.c)
   proof
    let c be object;
    assume
A4: c in dom aG; then
A5:c in dom(a(#)g);
A6:f/.c = g/.c by A1,REAL_NS1:def 4;
A7: f/.c = f.c by A2,A4,PARTFUN1:def 6;
    aG.c = (a(#)g)/.c by A4,PARTFUN1:def 6
    .= a*(g/.c) by A5,VFUNCT_1:def 4;
    hence aG.c = a(#)(f.c) by A6,A7,REAL_NS1:3;
   end;
   hence thesis by A2,A3,VALUED_2:def 39;
end;
