let f be constant standard special_circular_sequence; :: thesis: for g being FinSequence of (TOP-REAL 2)
for i1, i2 being Nat st g is_a_part_of f,i1,i2 holds
( 1 <= i1 & i1 + 1 <= len f & 1 <= i2 & i2 + 1 <= len f & g . (len g) = f . i2 & 1 <= len g & len g < len f & ( for i being Nat st 1 <= i & i <= len g holds
g . i = f . (S_Drop (((i1 + i) -' 1),f)) or for i being Nat st 1 <= i & i <= len g holds
g . i = f . (S_Drop ((((len f) + i1) -' i),f)) ) )
let g be FinSequence of (TOP-REAL 2); :: thesis: for i1, i2 being Nat st g is_a_part_of f,i1,i2 holds
( 1 <= i1 & i1 + 1 <= len f & 1 <= i2 & i2 + 1 <= len f & g . (len g) = f . i2 & 1 <= len g & len g < len f & ( for i being Nat st 1 <= i & i <= len g holds
g . i = f . (S_Drop (((i1 + i) -' 1),f)) or for i being Nat st 1 <= i & i <= len g holds
g . i = f . (S_Drop ((((len f) + i1) -' i),f)) ) )
let i1, i2 be Nat; :: thesis: ( g is_a_part_of f,i1,i2 implies ( 1 <= i1 & i1 + 1 <= len f & 1 <= i2 & i2 + 1 <= len f & g . (len g) = f . i2 & 1 <= len g & len g < len f & ( for i being Nat st 1 <= i & i <= len g holds
g . i = f . (S_Drop (((i1 + i) -' 1),f)) or for i being Nat st 1 <= i & i <= len g holds
g . i = f . (S_Drop ((((len f) + i1) -' i),f)) ) ) )
assume A1: g is_a_part_of f,i1,i2 ; :: thesis: ( 1 <= i1 & i1 + 1 <= len f & 1 <= i2 & i2 + 1 <= len f & g . (len g) = f . i2 & 1 <= len g & len g < len f & ( for i being Nat st 1 <= i & i <= len g holds
g . i = f . (S_Drop (((i1 + i) -' 1),f)) or for i being Nat st 1 <= i & i <= len g holds
g . i = f . (S_Drop ((((len f) + i1) -' i),f)) ) )

now :: thesis: ( ( g is_a_part>_of f,i1,i2 & 1 <= i1 & i1 + 1 <= len f & 1 <= i2 & i2 + 1 <= len f & g . (len g) = f . i2 & 1 <= len g & len g < len f & ( for i being Nat st 1 <= i & i <= len g holds
g . i = f . (S_Drop (((i1 + i) -' 1),f)) or for i being Nat st 1 <= i & i <= len g holds
g . i = f . (S_Drop ((((len f) + i1) -' i),f)) ) ) or ( g is_a_part<_of f,i1,i2 & 1 <= i1 & i1 + 1 <= len f & 1 <= i2 & i2 + 1 <= len f & g . (len g) = f . i2 & 1 <= len g & len g < len f & ( for i being Nat st 1 <= i & i <= len g holds
g . i = f . (S_Drop (((i1 + i) -' 1),f)) or for i being Nat st 1 <= i & i <= len g holds
g . i = f . (S_Drop ((((len f) + i1) -' i),f)) ) ) )

per cases ( g is_a_part>_of f,i1,i2 or g is_a_part<_of f,i1,i2 ) by A1;

case g is_a_part>_of f,i1,i2 ; :: thesis: ( 1 <= i1 & i1 + 1 <= len f & 1 <= i2 & i2 + 1 <= len f & g . (len g) = f . i2 & 1 <= len g & len g < len f & ( for i being Nat st 1 <= i & i <= len g holds
g . i = f . (S_Drop (((i1 + i) -' 1),f)) or for i being Nat st 1 <= i & i <= len g holds
g . i = f . (S_Drop ((((len f) + i1) -' i),f)) ) )

hence ( 1 <= i1 & i1 + 1 <= len f & 1 <= i2 & i2 + 1 <= len f & g . (len g) = f . i2 & 1 <= len g & len g < len f & ( for i being Nat st 1 <= i & i <= len g holds
g . i = f . (S_Drop (((i1 + i) -' 1),f)) or for i being Nat st 1 <= i & i <= len g holds
g . i = f . (S_Drop ((((len f) + i1) -' i),f)) ) ) ; :: thesis: verum

end;

case g is_a_part<_of f,i1,i2 ; :: thesis: ( 1 <= i1 & i1 + 1 <= len f & 1 <= i2 & i2 + 1 <= len f & g . (len g) = f . i2 & 1 <= len g & len g < len f & ( for i being Nat st 1 <= i & i <= len g holds
g . i = f . (S_Drop (((i1 + i) -' 1),f)) or for i being Nat st 1 <= i & i <= len g holds
g . i = f . (S_Drop ((((len f) + i1) -' i),f)) ) )

hence ( 1 <= i1 & i1 + 1 <= len f & 1 <= i2 & i2 + 1 <= len f & g . (len g) = f . i2 & 1 <= len g & len g < len f & ( for i being Nat st 1 <= i & i <= len g holds
g . i = f . (S_Drop (((i1 + i) -' 1),f)) or for i being Nat st 1 <= i & i <= len g holds
g . i = f . (S_Drop ((((len f) + i1) -' i),f)) ) ) ; :: thesis: verum

end;

end;

end;

hence ( 1 <= i1 & i1 + 1 <= len f & 1 <= i2 & i2 + 1 <= len f & g . (len g) = f . i2 & 1 <= len g & len g < len f & ( for i being Nat st 1 <= i & i <= len g holds
g . i = f . (S_Drop (((i1 + i) -' 1),f)) or for i being Nat st 1 <= i & i <= len g holds
g . i = f . (S_Drop ((((len f) + i1) -' i),f)) ) ) ; :: thesis: verum