let p be Element of CQC-WFF ; :: thesis: for x being bound_QC-variable
for Sub being CQC_Substitution 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; :: thesis: for Sub being CQC_Substitution 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; :: 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 by MCART_1:7;
A2: [(All (x,p)),Sub] `2 = Sub by MCART_1:7;
A3: [(All (x,p)),Sub] `1 = All (x,p) by MCART_1:7;
then ( 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, A3, A2, SUBSTUT1:def 36; :: thesis: verum