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 Th62:
  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} = {} & f2 is total;
    f2^ is total by A2,Th61;
    then f1(#)(f2^) is total by A1;
    hence thesis by Th38;
  end;
  assume f1/f2 is total;
  then
A3: f1(#)(f2^) is total by Th38;
  hence f1 is total by Th57;
  f2^ is total by A3,Th57;
  hence thesis by Th61;
end;
