let D be non empty set ; :: thesis: for f being FinSequence of D
for p1, p2, p3 being Element of D st p1 in rng f & p2 in rng f & p3 in rng f & p1 .. f < p2 .. f & p2 .. f <= p3 .. f holds
p2 .. (Rotate (f,p1)) <= p3 .. (Rotate (f,p1))

let f be FinSequence of D; :: thesis: for p1, p2, p3 being Element of D st p1 in rng f & p2 in rng f & p3 in rng f & p1 .. f < p2 .. f & p2 .. f <= p3 .. f holds
p2 .. (Rotate (f,p1)) <= p3 .. (Rotate (f,p1))

let p1, p2, p3 be Element of D; :: thesis: ( p1 in rng f & p2 in rng f & p3 in rng f & p1 .. f < p2 .. f & p2 .. f <= p3 .. f implies p2 .. (Rotate (f,p1)) <= p3 .. (Rotate (f,p1)) )
assume that
A1: p1 in rng f and
A2: ( p2 in rng f & p3 in rng f ) and
A3: p1 .. f < p2 .. f and
A4: p2 .. f <= p3 .. f ; :: thesis: p2 .. (Rotate (f,p1)) <= p3 .. (Rotate (f,p1))
per cases ( p2 .. f < p3 .. f or p2 .. f = p3 .. f ) by A4, XXREAL_0:1;
suppose p2 .. f < p3 .. f ; :: thesis: p2 .. (Rotate (f,p1)) <= p3 .. (Rotate (f,p1))
hence p2 .. (Rotate (f,p1)) <= p3 .. (Rotate (f,p1)) by A1, A2, A3, Th5; :: thesis: verum
end;
suppose p2 .. f = p3 .. f ; :: thesis: p2 .. (Rotate (f,p1)) <= p3 .. (Rotate (f,p1))
hence p2 .. (Rotate (f,p1)) <= p3 .. (Rotate (f,p1)) by A2, FINSEQ_5:9; :: thesis: verum
end;
end;