let f be FinSequence of (TOP-REAL 2); :: thesis: for Q being Subset of (TOP-REAL 2)
for q being Point of (TOP-REAL 2)
for i being Element of NAT st LSeg f,i meets Q & f is being_S-Seq & Q is closed & 1 <= i & i + 1 <= len f & q in LSeg f,i & q in Q holds
LE First_Point (LSeg f,i),(f /. i),(f /. (i + 1)),Q,q,f /. i,f /. (i + 1)
let Q be Subset of (TOP-REAL 2); :: thesis: for q being Point of (TOP-REAL 2)
for i being Element of NAT st LSeg f,i meets Q & f is being_S-Seq & Q is closed & 1 <= i & i + 1 <= len f & q in LSeg f,i & q in Q holds
LE First_Point (LSeg f,i),(f /. i),(f /. (i + 1)),Q,q,f /. i,f /. (i + 1)
let q be Point of (TOP-REAL 2); :: thesis: for i being Element of NAT st LSeg f,i meets Q & f is being_S-Seq & Q is closed & 1 <= i & i + 1 <= len f & q in LSeg f,i & q in Q holds
LE First_Point (LSeg f,i),(f /. i),(f /. (i + 1)),Q,q,f /. i,f /. (i + 1)
let i be Element of NAT ; :: thesis: ( LSeg f,i meets Q & f is being_S-Seq & Q is closed & 1 <= i & i + 1 <= len f & q in LSeg f,i & q in Q implies LE First_Point (LSeg f,i),(f /. i),(f /. (i + 1)),Q,q,f /. i,f /. (i + 1) )
assume that
A1: LSeg f,i meets Q and
A2: f is being_S-Seq and
A3: Q is closed and
A4: ( 1 <= i & i + 1 <= len f ) and
A5: q in LSeg f,i and
A6: q in Q ; :: thesis: LE First_Point (LSeg f,i),(f /. i),(f /. (i + 1)),Q,q,f /. i,f /. (i + 1)
reconsider P = LSeg f,i as non empty Subset of (TOP-REAL 2) by A5;
set q1 = First_Point P,(f /. i),(f /. (i + 1)),Q;
set p1 = f /. i;
set p2 = f /. (i + 1);
A7: P /\ Q c= P by XBOOLE_1:17;
A8: i + 1 in dom f by A4, SEQ_4:151;
A9: ( f is one-to-one & i in dom f ) by A2, A4, SEQ_4:151, TOPREAL1:def 10;
A10: f /. i <> f /. (i + 1)

proof

assume f /. i = f /. (i + 1) ; :: thesis: contradiction
then i = i + 1 by A9, A8, PARTFUN2:17;
hence contradiction ; :: thesis: verum

end;

A11: P /\ Q is closed by A3, TOPS_1:35;
P is_an_arc_of f /. i,f /. (i + 1) by A2, A4, JORDAN5B:15;
then ( First_Point P,(f /. i),(f /. (i + 1)),Q in P /\ Q & ( for g being Function of I[01] ,((TOP-REAL 2) | P)
for s1, s2 being Real st g is being_homeomorphism & g . 0 = f /. i & g . 1 = f /. (i + 1) & g . s1 = First_Point P,(f /. i),(f /. (i + 1)),Q & 0 <= s1 & s1 <= 1 & g . s2 = q & 0 <= s2 & s2 <= 1 holds
s1 <= s2 ) ) by A1, A6, A11, Def1;
then A12: LE First_Point P,(f /. i),(f /. (i + 1)),Q,q,P,f /. i,f /. (i + 1) by A5, A7, Def3;
LSeg f,i = LSeg (f /. i),(f /. (i + 1)) by A4, TOPREAL1:def 5;
hence LE First_Point (LSeg f,i),(f /. i),(f /. (i + 1)),Q,q,f /. i,f /. (i + 1) by A10, A12, Th17; :: thesis: verum