reserve x,y,X,Y for set;
reserve C for non empty set;
reserve c for Element of C;
reserve f,f1,f2,f3,g,g1 for PartFunc of C,COMPLEX;
reserve r1,r2,p1 for Real;
reserve r,q,cr1,cr2 for Complex;

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
A1: f1 is total & f2 is total;
  then f1+f2 is total;
  then dom (f1+f2) = C;
  hence (f1+f2)/.c = ((f1/.c)) + ((f2/.c)) by Th1;
  f1-f2 is total by A1;
  then dom (f1-f2) = C;
  hence (f1-f2)/.c = ((f1/.c)) - ((f2/.c)) by Th2;
  f1(#)f2 is total by A1;
  then dom (f1(#)f2) = C;
  hence thesis by Th3;
end;
