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 Th55:
  f1 is total & f2"{0} = {} & f2 is total iff f1/f2 is total
proof
  thus f1 is total & f2"{0} = {} & f2 is total implies f1/f2 is total
  proof
    assume that
A1: f1 is total and
A2: f2"{0} = {} and
A3: f2 is total;
    f2^ is total by A2,A3,Th54;
    then f1(#)(f2^) is total by A1;
    hence thesis by Th31;
  end;
  assume f1/f2 is total;
  then
A4: f1(#)(f2^) is total by Th31;
  hence f1 is total by Th50;
  f2^ is total by A4,Th50;
  hence thesis by Th54;
end;
