reserve n,m,i,k for Element of NAT;
reserve x,X,X1 for set;
reserve r,p for Real;
reserve s,x0,x1,x2 for Real;
reserve f,f1,f2 for PartFunc of REAL,REAL n;
reserve h for PartFunc of REAL,REAL-NS n;
reserve W for non empty set;

theorem Th6:
  for f1 be PartFunc of W,REAL-NS n,
      g1 be PartFunc of W,REAL n,
      a be Real st f1=g1 holds
    a(#)f1 = a(#)g1
proof
  let f1 be PartFunc of W,REAL-NS n,
      g1 be PartFunc of W,REAL n,
      a be Real;
  assume A1: f1=g1;
A2:  dom(a(#)f1) = dom f1 by VFUNCT_1:def 4; then
A3: dom(a(#)f1) = dom(a(#)g1) by A1,VALUED_2:def 39;
A4: now
    let x be Element of W;
    assume A5: x in dom(a(#)f1); then
A6:  g1.x = g1/.x by A1,A2,PARTFUN1:def 6;
    f1/.x=g1/.x by A1,REAL_NS1:def 4; then
A7: a*(f1/.x)=a*(g1/.x) by REAL_NS1:3;
A8: (a(#)f1)/.x = a*(f1/.x) by A5,VFUNCT_1:def 4;
    (a(#)g1).x = a(#)(g1.x) by A3,A5,VALUED_2:def 39;
    hence (a(#)f1).x = (a(#)g1).x by A5,A7,A8,A6,PARTFUN1:def 6;
  end;
  a(#)f1 is PartFunc of W,REAL n by REAL_NS1:def 4;
  hence thesis by A3,A4,PARTFUN1:5;
end;
