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

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

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

assume that

A1: f is being_S-Seq and

A2: ( 1 <= i & i + 1 <= len f ) ; :: thesis: LE f /. i,f /. (i + 1), L~ f,f /. 1,f /. (len f)

set p1 = f /. 1;

set p2 = f /. (len f);

set q1 = f /. i;

set q2 = f /. (i + 1);

A3: len f >= 2 by A1, TOPREAL1:def 8;

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

i + 1 in dom f by A2, SEQ_4:134;

then A4: f /. (i + 1) in P by A3, GOBOARD1:1;

A5: 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 = f /. (i + 1) & 0 <= s2 & s2 <= 1 holds

s1 <= s2

then f /. i in P by A3, GOBOARD1:1;

hence LE f /. i,f /. (i + 1), L~ f,f /. 1,f /. (len f) by A4, A5; :: thesis: verum

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

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

assume that

A1: f is being_S-Seq and

A2: ( 1 <= i & i + 1 <= len f ) ; :: thesis: LE f /. i,f /. (i + 1), L~ f,f /. 1,f /. (len f)

set p1 = f /. 1;

set p2 = f /. (len f);

set q1 = f /. i;

set q2 = f /. (i + 1);

A3: len f >= 2 by A1, TOPREAL1:def 8;

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

i + 1 in dom f by A2, SEQ_4:134;

then A4: f /. (i + 1) in P by A3, GOBOARD1:1;

A5: 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 = f /. (i + 1) & 0 <= s2 & s2 <= 1 holds

s1 <= s2

proof

i in dom f
by A2, SEQ_4:134;
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 = f /. (i + 1) & 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 = f /. (i + 1) & 0 <= s2 & s2 <= 1 implies s1 <= s2 )

assume that

A6: g is being_homeomorphism and

A7: ( g . 0 = f /. 1 & g . 1 = f /. (len f) ) and

A8: g . s1 = f /. i and

A9: ( 0 <= s1 & s1 <= 1 ) and

A10: g . s2 = f /. (i + 1) and

A11: ( 0 <= s2 & s2 <= 1 ) ; :: thesis: s1 <= s2

A12: dom g = [#] I[01] by A6, TOPS_2:def 5

.= the carrier of I[01] ;

then A13: s1 in dom g by A9, BORSUK_1:43;

A14: s2 in dom g by A11, A12, BORSUK_1:43;

A15: g is one-to-one by A6, TOPS_2:def 5;

consider r1, r2 being Real such that

A16: r1 < r2 and

A17: 0 <= r1 and

A18: r1 <= 1 and

0 <= r2 and

A19: r2 <= 1 and

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

A20: g . r1 = f /. i and

A21: g . r2 = f /. (i + 1) by A1, A2, A6, A7, JORDAN5B:7;

A22: r2 in dom g by A16, A17, A19, A12, BORSUK_1:43;

r1 in dom g by A17, A18, A12, BORSUK_1:43;

then s1 = r1 by A8, A20, A13, A15, FUNCT_1:def 4;

hence s1 <= s2 by A10, A16, A21, A22, A14, A15, 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 = f /. (i + 1) & 0 <= s2 & s2 <= 1 implies s1 <= s2 )

assume that

A6: g is being_homeomorphism and

A7: ( g . 0 = f /. 1 & g . 1 = f /. (len f) ) and

A8: g . s1 = f /. i and

A9: ( 0 <= s1 & s1 <= 1 ) and

A10: g . s2 = f /. (i + 1) and

A11: ( 0 <= s2 & s2 <= 1 ) ; :: thesis: s1 <= s2

A12: dom g = [#] I[01] by A6, TOPS_2:def 5

.= the carrier of I[01] ;

then A13: s1 in dom g by A9, BORSUK_1:43;

A14: s2 in dom g by A11, A12, BORSUK_1:43;

A15: g is one-to-one by A6, TOPS_2:def 5;

consider r1, r2 being Real such that

A16: r1 < r2 and

A17: 0 <= r1 and

A18: r1 <= 1 and

0 <= r2 and

A19: r2 <= 1 and

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

A20: g . r1 = f /. i and

A21: g . r2 = f /. (i + 1) by A1, A2, A6, A7, JORDAN5B:7;

A22: r2 in dom g by A16, A17, A19, A12, BORSUK_1:43;

r1 in dom g by A17, A18, A12, BORSUK_1:43;

then s1 = r1 by A8, A20, A13, A15, FUNCT_1:def 4;

hence s1 <= s2 by A10, A16, A21, A22, A14, A15, FUNCT_1:def 4; :: thesis: verum

then f /. i in P by A3, GOBOARD1:1;

hence LE f /. i,f /. (i + 1), L~ f,f /. 1,f /. (len f) by A4, A5; :: thesis: verum