let A be transitive RelStr ; :: thesis: for B, C being Subset of A
for s1 being FinSequence of A
for x being Element of A st s1 is C -desc_ordering & not x in C & B = C \/ {x} & ( for y being Element of A st y in C holds
x <= y ) holds
ex s2 being FinSequence of A st
( s2 = s1 ^ <*x*> & s2 is B -desc_ordering )

let B, C be Subset of A; :: thesis: for s1 being FinSequence of A
for x being Element of A st s1 is C -desc_ordering & not x in C & B = C \/ {x} & ( for y being Element of A st y in C holds
x <= y ) holds
ex s2 being FinSequence of A st
( s2 = s1 ^ <*x*> & s2 is B -desc_ordering )

let s1 be FinSequence of A; :: thesis: for x being Element of A st s1 is C -desc_ordering & not x in C & B = C \/ {x} & ( for y being Element of A st y in C holds
x <= y ) holds
ex s2 being FinSequence of A st
( s2 = s1 ^ <*x*> & s2 is B -desc_ordering )

let x be Element of A; :: thesis: ( s1 is C -desc_ordering & not x in C & B = C \/ {x} & ( for y being Element of A st y in C holds
x <= y ) implies ex s2 being FinSequence of A st
( s2 = s1 ^ <*x*> & s2 is B -desc_ordering ) )

assume that
A1: s1 is C -desc_ordering and
A2: ( not x in C & B = C \/ {x} & ( for y being Element of A st y in C holds
x <= y ) ) ; :: thesis: ex s2 being FinSequence of A st
( s2 = s1 ^ <*x*> & s2 is B -desc_ordering )

Rev (Rev s1) is C -desc_ordering by A1;
then Rev s1 is C -asc_ordering by Th75;
then consider s2 being FinSequence of A such that
A3: s2 = <*x*> ^ (Rev s1) and
A4: s2 is B -asc_ordering by ;
take Rev s2 ; :: thesis: ( Rev s2 = s1 ^ <*x*> & Rev s2 is B -desc_ordering )
thus Rev s2 = (Rev (Rev s1)) ^ () by
.= s1 ^ <*x*> by FINSEQ_5:60 ; :: thesis: Rev s2 is B -desc_ordering
thus Rev s2 is B -desc_ordering by ; :: thesis: verum