reserve i,n,m for Nat;

theorem Th8:
for f1,f2 be Function of REAL m,REAL n, g1,g2 be Point of
       R_NormSpace_of_BoundedLinearOperators(REAL-NS m,REAL-NS n) st
f1 = g1 & f2 = g2 holds f1 - f2 = g1 - g2
proof
   let f1,f2 be Function of REAL m,REAL n,
       g1,g2 be Point of
         R_NormSpace_of_BoundedLinearOperators(REAL-NS m,REAL-NS n);
   assume A1: f1 = g1 & f2 = g2;
A2:the carrier of REAL-NS m=REAL m
   & the carrier of REAL-NS n= REAL n by REAL_NS1:def 4; then
   reconsider g12 = g1 - g2 as Function of REAL m,REAL n by LOPBAN_1:def 9;
   g1 is LinearOperator of REAL-NS m,REAL-NS n &
   g2 is LinearOperator of REAL-NS m,REAL-NS n by LOPBAN_1:def 9; then
 dom g1 = REAL m & dom g2 = REAL m by A2,FUNCT_2:def 1; then
A3:dom f1 /\ dom f2 = dom g12 by A1,FUNCT_2:def 1;
A4: f1<-->f2 = f1-f2 by INTEGR15:def 10;
   for c be object st c in dom g12 holds g12.c = f1.c - f2.c
   proof
    let c be object;
    assume
A5: c in dom g12;
    then reconsider x = c as VECTOR of REAL-NS m by REAL_NS1:def 4;
    reconsider c1 = c as Element of REAL m by A5;
    g12.x = g1.x - g2.x by LOPBAN_1:40;
    hence g12.c = f1/.c1 - f2/.c1 by A1,REAL_NS1:5
    .= f1.c - f2.c;
   end;
   hence thesis by A3,A4,VALUED_2:def 46;
end;
