reserve i,n,m for Nat;

theorem Th9:
for f be Function of REAL m,REAL n,
    g be Point of R_NormSpace_of_BoundedLinearOperators(REAL-NS m,REAL-NS n),
    r be Real st
 f = g holds r(#)f = r*g
proof
   let f be Function of REAL m,REAL n,
       g be Point of
          R_NormSpace_of_BoundedLinearOperators(REAL-NS m,REAL-NS n),
       r 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 rG = r*g as Function of REAL m,REAL n by LOPBAN_1:def 9;
   dom f = REAL m by FUNCT_2:def 1; then
A2:dom f = dom rG by FUNCT_2:def 1;
A3: r(#)f = f[#]r by INTEGR15:def 11;
   for c be object st c in dom rG holds rG.c = r(#)f.c
   proof
    let c be object;
    assume
A4: c in dom rG;
    then reconsider x = c as VECTOR of REAL-NS m by REAL_NS1:def 4;
    reconsider c1 = c as Element of REAL m by A4;
    rG.x = r* g.x by LOPBAN_1:36;
    hence rG.c = r*f/.c1 by A1,REAL_NS1:3
    .= r(#)f.c;
   end;
   hence thesis by A2,A3,VALUED_2:def 39;
end;
