let D be non empty set ; :: thesis:
let p1, p2 be Element of D; :: thesis:
let f be FinSequence of D; :: thesis:

per cases ;

suppose A1: p1 in rng f ; :: thesis:

assume that
A2: p2 in rng (f /^ 1) and
A3: f just_once_values p2 ; :: thesis:
f = ((f -| p1) ^ <*p1*>) ^ (f |-- p1) by A1, FINSEQ_4:51;
then A4: p2 in (rng ((f -| p1) ^ <*p1*>)) \+\ (rng (f |-- p1)) by A3, Th16;

A5: now :: thesis:

per cases by A4, XBOOLE_0:1;

case that p2 in rng ((f -| p1) ^ <*p1*>) and
A6: not p2 in rng (f |-- p1) ; :: thesis:

now :: thesis:

per cases ;

case not p2 in rng <*p1*> ; :: thesis:

then not p2 in (rng (f |-- p1)) \/ (rng <*p1*>) by A6, XBOOLE_0:def 3;
then not p2 in rng (<*p1*> ^ (f |-- p1)) by FINSEQ_1:31;
hence not p2 in rng (f :- p1) by A1, Th41; :: thesis:

end;

case p2 in rng <*p1*> ; :: thesis:

then p2 in {p1} by FINSEQ_1:39;
hence p2 = p1 by TARSKI:def 1; :: thesis:

end;

end;

end;

hence ( p2 in rng (f :- p1) implies p1 = p2 ) ; :: thesis:

end;

case not p2 in rng ((f -| p1) ^ <*p1*>) ; :: thesis:

hence not p2 in rng (f -: p1) by A1, Th40; :: thesis:

end;

end;

end;

A7: p2 in rng f by A3, FINSEQ_4:5;
then f = <*(f /. 1)*> ^ (f /^ 1) by FINSEQ_5:29, RELAT_1:38;
then p2 in (rng <*(f /. 1)*>) \+\ (rng (f /^ 1)) by A3, Th16;
then not p2 in rng <*(f /. 1)*> by A2, XBOOLE_0:1;
then not p2 in {(f /. 1)} by FINSEQ_1:39;
then p2 <> f /. 1 by TARSKI:def 1;
then A8: p2 .. f <> 1 by A7, FINSEQ_5:38;

now :: thesis:

per cases by A7, A5, XBOOLE_0:def 5;

suppose p1 = p2 ; :: thesis:

hence Rotate ((Rotate (f,p1)),p2) = Rotate (f,p2) by Th93; :: thesis:

end;

suppose p2 in (rng f) \ (rng (f -: p1)) ; :: thesis:

hence Rotate ((Rotate (f,p1)),p2) = Rotate (f,p2) by Th102; :: thesis:

end;

suppose p2 in (rng f) \ (rng (f :- p1)) ; :: thesis:

hence Rotate ((Rotate (f,p1)),p2) = Rotate (f,p2) by A8, Th103; :: thesis:

end;

end;

end;

hence Rotate ((Rotate (f,p1)),p2) = Rotate (f,p2) ; :: thesis:

end;

suppose not p1 in rng f ; :: thesis:

hence ( p2 in rng (f /^ 1) & f just_once_values p2 implies Rotate ((Rotate (f,p1)),p2) = Rotate (f,p2) ) by Def2; :: thesis:

end;

end;