reserve x,y,X,Y for set;
reserve C,D,E for non empty set;
reserve SC for Subset of C;
reserve SD for Subset of D;
reserve SE for Subset of E;
reserve c,c1,c2 for Element of C;
reserve d,d1,d2 for Element of D;
reserve e for Element of E;
reserve f,f1,g for PartFunc of C,D;
reserve t for PartFunc of D,C;
reserve s for PartFunc of D,E;
reserve h for PartFunc of C,E;
reserve F for PartFunc of D,D;

theorem
  (for c1,c2 st c1 in dom f & c2 in dom f & f/.c1 = f/.c2 holds c1 = c2)
  implies f is one-to-one
proof
  assume
A1: for c1,c2 st c1 in dom f & c2 in dom f & f/.c1 = f/.c2 holds c1 = c2;
  now
    let x,y be object;
    assume that
A2: x in dom f and
A3: y in dom f and
A4: (f qua Function).x = (f qua Function).y;
    reconsider y1 = y as Element of C by A3;
    reconsider x1 = x as Element of C by A2;
    f/.x1 = (f qua Function).y1 by A2,A4,PARTFUN1:def 6;
    then f/.x1 = f/.y1 by A3,PARTFUN1:def 6;
    hence x=y by A1,A2,A3;
  end;
  hence thesis;
end;
