let f be FinSequence of (TOP-REAL 2); :: thesis: ( f is weakly-one-to-one implies for p, q being Point of (TOP-REAL 2) st p in L~ f & q in L~ f holds
B_Cut f,p,q = Rev (B_Cut f,q,p) )
assume A1: f is weakly-one-to-one ; :: thesis: for p, q being Point of (TOP-REAL 2) st p in L~ f & q in L~ f holds
B_Cut f,p,q = Rev (B_Cut f,q,p)
let p, q be Point of (TOP-REAL 2); :: thesis: ( p in L~ f & q in L~ f implies B_Cut f,p,q = Rev (B_Cut f,q,p) )
assume that
A2: p in L~ f and
A3: q in L~ f ; :: thesis: B_Cut f,p,q = Rev (B_Cut f,q,p)

per cases ( p = q or ( p <> q & ( Index p,f < Index q,f or ( Index p,f = Index q,f & LE p,q,f /. (Index p,f),f /. ((Index p,f) + 1) ) ) ) or ( p <> q & not Index p,f < Index q,f & not ( Index p,f = Index q,f & LE p,q,f /. (Index p,f),f /. ((Index p,f) + 1) ) ) ) ;

suppose A4: p = q ; :: thesis: B_Cut f,p,q = Rev (B_Cut f,q,p)

then B_Cut f,p,q = <*p*> by A1, A2, Th15;
hence B_Cut f,p,q = Rev (B_Cut f,q,p) by A4, FINSEQ_5:63; :: thesis: verum

end;

suppose that A5: p <> q and
A6: ( Index p,f < Index q,f or ( Index p,f = Index q,f & LE p,q,f /. (Index p,f),f /. ((Index p,f) + 1) ) ) ; :: thesis: B_Cut f,p,q = Rev (B_Cut f,q,p)

( not Index q,f = Index p,f or not LE q,p,f /. (Index q,f),f /. ((Index q,f) + 1) ) by A5, A6, JORDAN3:62;
then A7: Rev (B_Cut f,q,p) = Rev (Rev (R_Cut (L_Cut f,p),q)) by A6, JORDAN3:def 8;
B_Cut f,p,q = R_Cut (L_Cut f,p),q by A2, A3, A6, JORDAN3:def 8;
hence B_Cut f,p,q = Rev (B_Cut f,q,p) by A7, FINSEQ_6:29; :: thesis: verum

end;

suppose that p <> q and
A8: ( not Index p,f < Index q,f & not ( Index p,f = Index q,f & LE p,q,f /. (Index p,f),f /. ((Index p,f) + 1) ) ) ; :: thesis: B_Cut f,p,q = Rev (B_Cut f,q,p)

A9: ( Index q,f < Index p,f or ( Index p,f = Index q,f & not LE p,q,f /. (Index p,f),f /. ((Index p,f) + 1) ) ) by A8, XXREAL_0:1;

A10: now

assume that
A11: Index p,f = Index q,f and
A12: not LE p,q,f /. (Index p,f),f /. ((Index p,f) + 1) ; :: thesis: LE q,p,f /. (Index q,f),f /. ((Index q,f) + 1)
A13: 1 <= Index p,f by A2, JORDAN3:41;
A14: Index p,f < len f by A2, JORDAN3:41;
then A15: (Index p,f) + 1 <= len f by NAT_1:13;
then A16: LSeg f,(Index p,f) = LSeg (f /. (Index p,f)),(f /. ((Index p,f) + 1)) by A13, TOPREAL1:def 5;
then A17: p in LSeg (f /. (Index p,f)),(f /. ((Index p,f) + 1)) by A2, JORDAN3:42;
1 <= (Index p,f) + 1 by NAT_1:11;
then A18: (Index p,f) + 1 in dom f by A15, FINSEQ_3:27;
f . (Index p,f) <> f . ((Index p,f) + 1) by A1, A13, A14, Def2;
then A19: f . (Index p,f) <> f /. ((Index p,f) + 1) by A18, PARTFUN1:def 8;
Index p,f in dom f by A13, A14, FINSEQ_3:27;
then A20: f /. (Index p,f) <> f /. ((Index p,f) + 1) by A19, PARTFUN1:def 8;
q in LSeg (f /. (Index p,f)),(f /. ((Index p,f) + 1)) by A3, A11, A16, JORDAN3:42;
then LT q,p,f /. (Index p,f),f /. ((Index p,f) + 1) by A12, A17, A20, JORDAN3:63;
hence LE q,p,f /. (Index q,f),f /. ((Index q,f) + 1) by A11, JORDAN3:def 7; :: thesis: verum

end;

B_Cut f,p,q = Rev (R_Cut (L_Cut f,q),p) by A8, JORDAN3:def 8;
hence B_Cut f,p,q = Rev (B_Cut f,q,p) by A2, A3, A9, A10, JORDAN3:def 8; :: thesis: verum

end;

end;