reserve x for object, X,Y for set;
reserve C for non empty set;
reserve c for Element of C;
reserve f,f1,f2,f3,g,g1 for complex-valued Function;
reserve r,p for Complex;
reserve r,r1,r2,p for Real;
reserve f,f1,f2 for PartFunc of C,REAL;

theorem
  f1 is total & f2 is total implies (f1+f2).c = f1.c + f2.c & (f1-f2).c
  = f1.c - f2.c & (f1(#)f2).c = f1.c * f2.c
proof
  assume that
A1: f1 is total and
A2: f2 is total;
  f1+f2 is total by A1,A2;
  then dom (f1+f2) = C;
  hence (f1+f2).c = f1.c + f2.c by VALUED_1:def 1;
  f1-f2 is total by A1,A2;
  then dom (f1-f2) = C;
  hence (f1-f2).c = f1.c - f2.c by VALUED_1:13;
  thus thesis by VALUED_1:5;
end;
