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 & i1 <> i2 holds
L~ g is_S-P_arc_joining f /. i1,f /. i2
let g be FinSequence of (TOP-REAL 2); :: thesis: for i1, i2 being Nat st g is_a_part<_of f,i1,i2 & i1 <> i2 holds
L~ g is_S-P_arc_joining f /. i1,f /. i2
let i1, i2 be Nat; :: thesis: ( g is_a_part<_of f,i1,i2 & i1 <> i2 implies L~ g is_S-P_arc_joining f /. i1,f /. i2 )
assume that
A1: g is_a_part<_of f,i1,i2 and
A2: i1 <> i2 ; :: thesis: L~ g is_S-P_arc_joining f /. i1,f /. i2

now :: thesis: ( ( i1 > i2 & L~ g is_S-P_arc_joining f /. i1,f /. i2 ) or ( i1 <= i2 & L~ g is_S-P_arc_joining f /. i1,f /. i2 ) )

per cases ( i1 > i2 or i1 <= i2 ) ;

case A3: i1 > i2 ; :: thesis: L~ g is_S-P_arc_joining f /. i1,f /. i2

reconsider P = L~ g as Subset of (TOP-REAL 2) ;
reconsider p2 = f /. i2, p1 = f /. i1 as Point of (TOP-REAL 2) ;
L~ (Rev g) is_S-P_arc_joining f /. i2,f /. i1 by A1, A3, Th30, Th44;
then P is_S-P_arc_joining p2,p1 by SPPOL_2:22;
hence L~ g is_S-P_arc_joining f /. i1,f /. i2 by SPPOL_2:49; :: thesis: verum

end;

case A4: i1 <= i2 ; :: thesis: L~ g is_S-P_arc_joining f /. i1,f /. i2

reconsider P = L~ g as Subset of (TOP-REAL 2) ;
reconsider p2 = f /. i2, p1 = f /. i1 as Point of (TOP-REAL 2) ;
i1 < i2 by A2, A4, XXREAL_0:1;
then L~ (Rev g) is_S-P_arc_joining f /. i2,f /. i1 by A1, Lm1, Th30;
then P is_S-P_arc_joining p2,p1 by SPPOL_2:22;
hence L~ g is_S-P_arc_joining f /. i1,f /. i2 by SPPOL_2:49; :: thesis: verum

end;

end;

end;

hence L~ g is_S-P_arc_joining f /. i1,f /. i2 ; :: thesis: verum