reserve x,r,a,x0,p for Real;
reserve n,i,m for Element of NAT;
reserve Z for open Subset of REAL;
reserve f,f1,f2 for PartFunc of REAL,REAL;
reserve k for Nat;

theorem Th5:
  for n,m be Real holds n(#)f1+m(#)f1=(n+m)(#)f1
proof
  let n,m be Real;
A1: dom(n(#)f1+m(#)f1) = dom (n(#)f1)/\dom (m(#)f1) by VALUED_1:def 1
    .= dom f1/\dom (m(#)f1)by VALUED_1:def 5
    .= dom f1/\dom f1 by VALUED_1:def 5
    .=dom f1;
A2: for x being Element of REAL st x in dom f1
      holds (n(#)f1+m(#)f1).x=((n+m)(#)f1).x
  proof
    let x be Element of REAL;
    assume
A3: x in dom f1;
    then
A4: x in dom(n(#)f1) by VALUED_1:def 5;
    x in dom((n+m)(#)f1) by A3,VALUED_1:def 5;
    then
A5: ((n+m)(#)f1).x = (n+m)*f1.x by VALUED_1:def 5
      .=n*f1.x +m*f1.x;
A6: x in dom(m(#)f1) by A3,VALUED_1:def 5;
    (n(#)f1+m(#)f1).x = (n(#)f1).x+(m(#)f1).x by A1,A3,VALUED_1:def 1
      .=n*f1.x +(m(#)f1).x by A4,VALUED_1:def 5
      .=n*f1.x +m*f1.x by A6,VALUED_1:def 5;
    hence thesis by A5;
  end;
  dom(n(#)f1+m(#)f1) = dom((n+m)(#)f1) by A1,VALUED_1:def 5;
  hence thesis by A1,A2,PARTFUN1:5;
end;
