reserve x,X,Y for set;
reserve C for non empty set;
reserve c for Element of C;
reserve V for RealNormSpace;
reserve f,f1,f2,f3 for PartFunc of C,V;
reserve r,r1,r2,p for Real;

theorem
  f1 is total & f2 is total implies (f1+f2)/.c = (f1/.c) + (f2/.c) &
  (f1-f2)/.c = (f1/.c) - (f2/.c)
proof
  assume
A1: f1 is total & f2 is total;
  then dom (f1+f2) = C by PARTFUN1:def 2;
  hence (f1+f2)/.c = (f1/.c) + (f2/.c) by Def1;
  dom (f1-f2) = C by A1,PARTFUN1:def 2;
  hence thesis by Def2;
end;
