let Al be QC-alphabet ; :: thesis: for p being Element of CQC-WFF Al

for x being bound_QC-variable of Al

for Sub being CQC_Substitution of Al st x in rng (RestrictSub (x,(All (x,p)),Sub)) holds

S_Bound [(All (x,p)),Sub] = x. (upVar ((RestrictSub (x,(All (x,p)),Sub)),p))

let p be Element of CQC-WFF Al; :: thesis: for x being bound_QC-variable of Al

for Sub being CQC_Substitution of Al st x in rng (RestrictSub (x,(All (x,p)),Sub)) holds

S_Bound [(All (x,p)),Sub] = x. (upVar ((RestrictSub (x,(All (x,p)),Sub)),p))

let x be bound_QC-variable of Al; :: thesis: for Sub being CQC_Substitution of Al st x in rng (RestrictSub (x,(All (x,p)),Sub)) holds

S_Bound [(All (x,p)),Sub] = x. (upVar ((RestrictSub (x,(All (x,p)),Sub)),p))

let Sub be CQC_Substitution of Al; :: thesis: ( x in rng (RestrictSub (x,(All (x,p)),Sub)) implies S_Bound [(All (x,p)),Sub] = x. (upVar ((RestrictSub (x,(All (x,p)),Sub)),p)) )

set finSub = RestrictSub (x,(All (x,p)),Sub);

set S = [(All (x,p)),Sub];

assume A1: x in rng (RestrictSub (x,(All (x,p)),Sub)) ; :: thesis: S_Bound [(All (x,p)),Sub] = x. (upVar ((RestrictSub (x,(All (x,p)),Sub)),p))

reconsider q = [(All (x,p)),Sub] `1 as Element of CQC-WFF Al ;

A2: [(All (x,p)),Sub] `2 = Sub ;

( bound_in q = x & the_scope_of q = p ) by QC_LANG2:7;

hence S_Bound [(All (x,p)),Sub] = x. (upVar ((RestrictSub (x,(All (x,p)),Sub)),p)) by A1, A2, SUBSTUT1:def 36; :: thesis: verum

for x being bound_QC-variable of Al

for Sub being CQC_Substitution of Al st x in rng (RestrictSub (x,(All (x,p)),Sub)) holds

S_Bound [(All (x,p)),Sub] = x. (upVar ((RestrictSub (x,(All (x,p)),Sub)),p))

let p be Element of CQC-WFF Al; :: thesis: for x being bound_QC-variable of Al

for Sub being CQC_Substitution of Al st x in rng (RestrictSub (x,(All (x,p)),Sub)) holds

S_Bound [(All (x,p)),Sub] = x. (upVar ((RestrictSub (x,(All (x,p)),Sub)),p))

let x be bound_QC-variable of Al; :: thesis: for Sub being CQC_Substitution of Al st x in rng (RestrictSub (x,(All (x,p)),Sub)) holds

S_Bound [(All (x,p)),Sub] = x. (upVar ((RestrictSub (x,(All (x,p)),Sub)),p))

let Sub be CQC_Substitution of Al; :: thesis: ( x in rng (RestrictSub (x,(All (x,p)),Sub)) implies S_Bound [(All (x,p)),Sub] = x. (upVar ((RestrictSub (x,(All (x,p)),Sub)),p)) )

set finSub = RestrictSub (x,(All (x,p)),Sub);

set S = [(All (x,p)),Sub];

assume A1: x in rng (RestrictSub (x,(All (x,p)),Sub)) ; :: thesis: S_Bound [(All (x,p)),Sub] = x. (upVar ((RestrictSub (x,(All (x,p)),Sub)),p))

reconsider q = [(All (x,p)),Sub] `1 as Element of CQC-WFF Al ;

A2: [(All (x,p)),Sub] `2 = Sub ;

( bound_in q = x & the_scope_of q = p ) by QC_LANG2:7;

hence S_Bound [(All (x,p)),Sub] = x. (upVar ((RestrictSub (x,(All (x,p)),Sub)),p)) by A1, A2, SUBSTUT1:def 36; :: thesis: verum