reserve a,b,p,x,x9,x1,x19,x2,y,y9,y1,y19,y2,z,z9,z1,z2 for object,
   X,X9,Y,Y9,Z,Z9 for set;
reserve A,D,D9 for non empty set;
reserve f,g,h for Function;
reserve A,B for set;
reserve x,y,i,j,k for object;

theorem Th100:
 for x,y being object holds
  x <> y implies not x in rng(f+~(x,y))
proof let x,y be object;
  assume
A1: x <> y;
  assume x in rng(f+~(x,y));
  then consider z being object such that
A2: z in dom(f+~(x,y)) and
A3: (f+~(x,y)).z = x by FUNCT_1:def 3;
A4: z in dom f by A2,Th99;
A5: now
    assume
A6: not z in dom((x .--> y)*f);
    then f.z = x by A3,Th11;
    then f.z in dom(x .--> y) by FUNCOP_1:74;
    hence contradiction by A4,A6,FUNCT_1:11;
  end;
  (x .--> y).(f.z) = ((x .--> y)*f).z by A4,FUNCT_1:13
    .= x by A3,A5,Th13;
  then f.z <> x by A1,FUNCOP_1:72;
  then not f.z in dom(x .--> y) by FUNCOP_1:75;
  hence thesis by A5,FUNCT_1:11;
end;
