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
  f is one-to-one & x in dom f & y in dom f & f/.x = f/.y implies x = y
proof
  assume that
A1: f is one-to-one and
A2: x in dom f and
A3: y in dom f;
  assume f/.x = f/.y;
  then f.x = f/.y by A2,PARTFUN1:def 6
    .= f.y by A3,PARTFUN1:def 6;
  hence thesis by A1,A2,A3;
end;
