let f be FinSequence of (TOP-REAL 2); :: thesis: for q being Point of (TOP-REAL 2)

for i being Nat st f is being_S-Seq & 1 <= i & i + 1 <= len f & q in LSeg (f,i) holds

LE f /. i,q, L~ f,f /. 1,f /. (len f)

let q be Point of (TOP-REAL 2); :: thesis: for i being Nat st f is being_S-Seq & 1 <= i & i + 1 <= len f & q in LSeg (f,i) holds

LE f /. i,q, L~ f,f /. 1,f /. (len f)

let i be Nat; :: thesis: ( f is being_S-Seq & 1 <= i & i + 1 <= len f & q in LSeg (f,i) implies LE f /. i,q, L~ f,f /. 1,f /. (len f) )

assume that

A1: f is being_S-Seq and

A2: ( 1 <= i & i + 1 <= len f ) and

A3: q in LSeg (f,i) ; :: thesis: LE f /. i,q, L~ f,f /. 1,f /. (len f)

set p1 = f /. 1;

set p2 = f /. (len f);

set q1 = f /. i;

A4: 2 <= len f by A1, TOPREAL1:def 8;

then reconsider P = L~ f as non empty Subset of (TOP-REAL 2) by TOPREAL1:23;

i in dom f by A2, SEQ_4:134;

then A5: f /. i in P by A4, GOBOARD1:1;

A6: for g being Function of I[01],((TOP-REAL 2) | P)

for s1, s2 being Real st g is being_homeomorphism & g . 0 = f /. 1 & g . 1 = f /. (len f) & g . s1 = f /. i & 0 <= s1 & s1 <= 1 & g . s2 = q & 0 <= s2 & s2 <= 1 holds

s1 <= s2

hence LE f /. i,q, L~ f,f /. 1,f /. (len f) by A5, A6; :: thesis: verum

for i being Nat st f is being_S-Seq & 1 <= i & i + 1 <= len f & q in LSeg (f,i) holds

LE f /. i,q, L~ f,f /. 1,f /. (len f)

let q be Point of (TOP-REAL 2); :: thesis: for i being Nat st f is being_S-Seq & 1 <= i & i + 1 <= len f & q in LSeg (f,i) holds

LE f /. i,q, L~ f,f /. 1,f /. (len f)

let i be Nat; :: thesis: ( f is being_S-Seq & 1 <= i & i + 1 <= len f & q in LSeg (f,i) implies LE f /. i,q, L~ f,f /. 1,f /. (len f) )

assume that

A1: f is being_S-Seq and

A2: ( 1 <= i & i + 1 <= len f ) and

A3: q in LSeg (f,i) ; :: thesis: LE f /. i,q, L~ f,f /. 1,f /. (len f)

set p1 = f /. 1;

set p2 = f /. (len f);

set q1 = f /. i;

A4: 2 <= len f by A1, TOPREAL1:def 8;

then reconsider P = L~ f as non empty Subset of (TOP-REAL 2) by TOPREAL1:23;

i in dom f by A2, SEQ_4:134;

then A5: f /. i in P by A4, GOBOARD1:1;

A6: for g being Function of I[01],((TOP-REAL 2) | P)

for s1, s2 being Real st g is being_homeomorphism & g . 0 = f /. 1 & g . 1 = f /. (len f) & g . s1 = f /. i & 0 <= s1 & s1 <= 1 & g . s2 = q & 0 <= s2 & s2 <= 1 holds

s1 <= s2

proof

q in P
by A3, SPPOL_2:17;
let g be Function of I[01],((TOP-REAL 2) | P); :: thesis: for s1, s2 being Real st g is being_homeomorphism & g . 0 = f /. 1 & g . 1 = f /. (len f) & g . s1 = f /. i & 0 <= s1 & s1 <= 1 & g . s2 = q & 0 <= s2 & s2 <= 1 holds

s1 <= s2

let s1, s2 be Real; :: thesis: ( g is being_homeomorphism & g . 0 = f /. 1 & g . 1 = f /. (len f) & g . s1 = f /. i & 0 <= s1 & s1 <= 1 & g . s2 = q & 0 <= s2 & s2 <= 1 implies s1 <= s2 )

assume that

A7: g is being_homeomorphism and

A8: ( g . 0 = f /. 1 & g . 1 = f /. (len f) ) and

A9: g . s1 = f /. i and

A10: ( 0 <= s1 & s1 <= 1 ) and

A11: g . s2 = q and

A12: ( 0 <= s2 & s2 <= 1 ) ; :: thesis: s1 <= s2

A13: dom g = [#] I[01] by A7, TOPS_2:def 5

.= the carrier of I[01] ;

then A14: s1 in dom g by A10, BORSUK_1:43;

consider r1, r2 being Real such that

r1 < r2 and

A15: ( 0 <= r1 & r1 <= 1 ) and

0 <= r2 and

r2 <= 1 and

A16: LSeg (f,i) = g .: [.r1,r2.] and

A17: g . r1 = f /. i and

g . r2 = f /. (i + 1) by A1, A2, A7, A8, JORDAN5B:7;

consider r39 being object such that

A18: r39 in dom g and

A19: r39 in [.r1,r2.] and

A20: g . r39 = q by A3, A16, FUNCT_1:def 6;

r39 in { l where l is Real : ( r1 <= l & l <= r2 ) } by A19, RCOMP_1:def 1;

then consider r3 being Real such that

A21: r3 = r39 and

A22: r1 <= r3 and

r3 <= r2 ;

A23: g is one-to-one by A7, TOPS_2:def 5;

A24: r1 in dom g by A15, A13, BORSUK_1:43;

s2 in dom g by A12, A13, BORSUK_1:43;

then s2 = r3 by A11, A18, A20, A21, A23, FUNCT_1:def 4;

hence s1 <= s2 by A9, A17, A22, A24, A14, A23, FUNCT_1:def 4; :: thesis: verum

end;s1 <= s2

let s1, s2 be Real; :: thesis: ( g is being_homeomorphism & g . 0 = f /. 1 & g . 1 = f /. (len f) & g . s1 = f /. i & 0 <= s1 & s1 <= 1 & g . s2 = q & 0 <= s2 & s2 <= 1 implies s1 <= s2 )

assume that

A7: g is being_homeomorphism and

A8: ( g . 0 = f /. 1 & g . 1 = f /. (len f) ) and

A9: g . s1 = f /. i and

A10: ( 0 <= s1 & s1 <= 1 ) and

A11: g . s2 = q and

A12: ( 0 <= s2 & s2 <= 1 ) ; :: thesis: s1 <= s2

A13: dom g = [#] I[01] by A7, TOPS_2:def 5

.= the carrier of I[01] ;

then A14: s1 in dom g by A10, BORSUK_1:43;

consider r1, r2 being Real such that

r1 < r2 and

A15: ( 0 <= r1 & r1 <= 1 ) and

0 <= r2 and

r2 <= 1 and

A16: LSeg (f,i) = g .: [.r1,r2.] and

A17: g . r1 = f /. i and

g . r2 = f /. (i + 1) by A1, A2, A7, A8, JORDAN5B:7;

consider r39 being object such that

A18: r39 in dom g and

A19: r39 in [.r1,r2.] and

A20: g . r39 = q by A3, A16, FUNCT_1:def 6;

r39 in { l where l is Real : ( r1 <= l & l <= r2 ) } by A19, RCOMP_1:def 1;

then consider r3 being Real such that

A21: r3 = r39 and

A22: r1 <= r3 and

r3 <= r2 ;

A23: g is one-to-one by A7, TOPS_2:def 5;

A24: r1 in dom g by A15, A13, BORSUK_1:43;

s2 in dom g by A12, A13, BORSUK_1:43;

then s2 = r3 by A11, A18, A20, A21, A23, FUNCT_1:def 4;

hence s1 <= s2 by A9, A17, A22, A24, A14, A23, FUNCT_1:def 4; :: thesis: verum

hence LE f /. i,q, L~ f,f /. 1,f /. (len f) by A5, A6; :: thesis: verum