consider f being Function of (TS SCM-AE ),(Funcs NAT ,(the Instructions of SCM ^omega )) such that
A1: ( ( for t being Terminal of SCM-AE ex g being Function of NAT ,(the Instructions of SCM ^omega ) st
( g = f . (root-tree t) & ( for n being Element of NAT holds g . n = H₃(t,n) ) ) ) & ( for nt being NonTerminal of SCM-AE
for t1, t2 being bin-term
for rtl, rtr being Symbol of SCM-AE st rtl = root-label t1 & rtr = root-label t2 & nt ==> <*rtl,rtr*> holds
ex g, f1, f2 being Function of NAT ,(the Instructions of SCM ^omega ) st
( g = f . (nt -tree t1,t2) & f1 = f . t1 & f2 = f . t2 & ( for n being Element of NAT holds g . n = H₂(nt,f1,f2,n) ) ) ) ) from BINTREE1:sch 5();
take f ; :: thesis: ( ( for t being Terminal of SCM-AE ex g being Function of NAT ,(the Instructions of SCM ^omega ) st
( g = f . (root-tree t) & ( for n being Element of NAT holds g . n = <%((dl. n) := (@ t))%> ) ) ) & ( for nt being NonTerminal of SCM-AE
for t1, t2 being bin-term
for rtl, rtr being Symbol of SCM-AE st rtl = root-label t1 & rtr = root-label t2 & nt ==> <*rtl,rtr*> holds
ex g, f1, f2 being Function of NAT ,(the Instructions of SCM ^omega ) st
( g = f . (nt -tree t1,t2) & f1 = f . t1 & f2 = f . t2 & ( for n being Element of NAT holds g . n = ((f1 . n) ^ (f2 . (n + 1))) ^ (Selfwork nt,n) ) ) ) )
( ( for t being Terminal of SCM-AE ex g being Function of NAT ,(the Instructions of SCM ^omega ) st
( g = f . (root-tree t) & ( for n being Element of NAT holds g . n = <%((dl. n) := (@ t))%> ) ) ) & ( for nt being NonTerminal of SCM-AE
for t1, t2 being bin-term
for rtl, rtr being Symbol of SCM-AE st rtl = root-label t1 & rtr = root-label t2 & nt ==> <*rtl,rtr*> holds
ex g, f1, f2 being Function of NAT ,(the Instructions of SCM ^omega ) st
( g = f . (nt -tree t1,t2) & f1 = f . t1 & f2 = f . t2 & ( for n being Element of NAT holds g . n = ((f1 . n) ^ (f2 . (n + 1))) ^ (Selfwork nt,n) ) ) ) )

proof

thus for t being Terminal of SCM-AE ex g being Function of NAT ,(the Instructions of SCM ^omega ) st
( g = f . (root-tree t) & ( for n being Element of NAT holds g . n = <%((dl. n) := (@ t))%> ) ) :: thesis: for nt being NonTerminal of SCM-AE
for t1, t2 being bin-term
for rtl, rtr being Symbol of SCM-AE st rtl = root-label t1 & rtr = root-label t2 & nt ==> <*rtl,rtr*> holds
ex g, f1, f2 being Function of NAT ,(the Instructions of SCM ^omega ) st
( g = f . (nt -tree t1,t2) & f1 = f . t1 & f2 = f . t2 & ( for n being Element of NAT holds g . n = ((f1 . n) ^ (f2 . (n + 1))) ^ (Selfwork nt,n) ) )

proof

let t be Terminal of SCM-AE ; :: thesis: ex g being Function of NAT ,(the Instructions of SCM ^omega ) st
( g = f . (root-tree t) & ( for n being Element of NAT holds g . n = <%((dl. n) := (@ t))%> ) )
consider g being Function of NAT ,(the Instructions of SCM ^omega ) such that
W1: g = f . (root-tree t) and
W2: for n being Element of NAT holds g . n = H₃(t,n) by A1;
take g ; :: thesis: ( g = f . (root-tree t) & ( for n being Element of NAT holds g . n = <%((dl. n) := (@ t))%> ) )
thus g = f . (root-tree t) by W1; :: thesis: for n being Element of NAT holds g . n = <%((dl. n) := (@ t))%>
let n be Element of NAT ; :: thesis: g . n = <%((dl. n) := (@ t))%>
g . n = H₃(t,n) by W2;
hence g . n = <%((dl. n) := (@ t))%> ; :: thesis: verum

end;

thus for nt being NonTerminal of SCM-AE
for t1, t2 being bin-term
for rtl, rtr being Symbol of SCM-AE st rtl = root-label t1 & rtr = root-label t2 & nt ==> <*rtl,rtr*> holds
ex g, f1, f2 being Function of NAT ,(the Instructions of SCM ^omega ) st
( g = f . (nt -tree t1,t2) & f1 = f . t1 & f2 = f . t2 & ( for n being Element of NAT holds g . n = ((f1 . n) ^ (f2 . (n + 1))) ^ (Selfwork nt,n) ) ) :: thesis: verum

proof

let nt be NonTerminal of SCM-AE ; :: thesis: for t1, t2 being bin-term
for rtl, rtr being Symbol of SCM-AE st rtl = root-label t1 & rtr = root-label t2 & nt ==> <*rtl,rtr*> holds
ex g, f1, f2 being Function of NAT ,(the Instructions of SCM ^omega ) st
( g = f . (nt -tree t1,t2) & f1 = f . t1 & f2 = f . t2 & ( for n being Element of NAT holds g . n = ((f1 . n) ^ (f2 . (n + 1))) ^ (Selfwork nt,n) ) )
let t1, t2 be bin-term; :: thesis: for rtl, rtr being Symbol of SCM-AE st rtl = root-label t1 & rtr = root-label t2 & nt ==> <*rtl,rtr*> holds
ex g, f1, f2 being Function of NAT ,(the Instructions of SCM ^omega ) st
( g = f . (nt -tree t1,t2) & f1 = f . t1 & f2 = f . t2 & ( for n being Element of NAT holds g . n = ((f1 . n) ^ (f2 . (n + 1))) ^ (Selfwork nt,n) ) )
let rtl, rtr be Symbol of SCM-AE ; :: thesis: ( rtl = root-label t1 & rtr = root-label t2 & nt ==> <*rtl,rtr*> implies ex g, f1, f2 being Function of NAT ,(the Instructions of SCM ^omega ) st
( g = f . (nt -tree t1,t2) & f1 = f . t1 & f2 = f . t2 & ( for n being Element of NAT holds g . n = ((f1 . n) ^ (f2 . (n + 1))) ^ (Selfwork nt,n) ) ) )
assume Z: ( rtl = root-label t1 & rtr = root-label t2 & nt ==> <*rtl,rtr*> ) ; :: thesis: ex g, f1, f2 being Function of NAT ,(the Instructions of SCM ^omega ) st
( g = f . (nt -tree t1,t2) & f1 = f . t1 & f2 = f . t2 & ( for n being Element of NAT holds g . n = ((f1 . n) ^ (f2 . (n + 1))) ^ (Selfwork nt,n) ) )
consider g, f1, f2 being Function of NAT ,(the Instructions of SCM ^omega ) such that
W1: ( g = f . (nt -tree t1,t2) & f1 = f . t1 & f2 = f . t2 ) and
W2: for n being Element of NAT holds g . n = H₂(nt,f1,f2,n) by Z, A1;
take g ; :: thesis: ex f1, f2 being Function of NAT ,(the Instructions of SCM ^omega ) st
( g = f . (nt -tree t1,t2) & f1 = f . t1 & f2 = f . t2 & ( for n being Element of NAT holds g . n = ((f1 . n) ^ (f2 . (n + 1))) ^ (Selfwork nt,n) ) )
take f1 ; :: thesis: ex f2 being Function of NAT ,(the Instructions of SCM ^omega ) st
( g = f . (nt -tree t1,t2) & f1 = f . t1 & f2 = f . t2 & ( for n being Element of NAT holds g . n = ((f1 . n) ^ (f2 . (n + 1))) ^ (Selfwork nt,n) ) )
take f2 ; :: thesis: ( g = f . (nt -tree t1,t2) & f1 = f . t1 & f2 = f . t2 & ( for n being Element of NAT holds g . n = ((f1 . n) ^ (f2 . (n + 1))) ^ (Selfwork nt,n) ) )
thus ( g = f . (nt -tree t1,t2) & f1 = f . t1 & f2 = f . t2 ) by W1; :: thesis: for n being Element of NAT holds g . n = ((f1 . n) ^ (f2 . (n + 1))) ^ (Selfwork nt,n)
let n be Element of NAT ; :: thesis: g . n = ((f1 . n) ^ (f2 . (n + 1))) ^ (Selfwork nt,n)
g . n = H₂(nt,f1,f2,n) by W2;
hence g . n = ((f1 . n) ^ (f2 . (n + 1))) ^ (Selfwork nt,n) ; :: thesis: verum

end;

end;

hence ( ( for t being Terminal of SCM-AE ex g being Function of NAT ,(the Instructions of SCM ^omega ) st
( g = f . (root-tree t) & ( for n being Element of NAT holds g . n = <%((dl. n) := (@ t))%> ) ) ) & ( for nt being NonTerminal of SCM-AE
for t1, t2 being bin-term
for rtl, rtr being Symbol of SCM-AE st rtl = root-label t1 & rtr = root-label t2 & nt ==> <*rtl,rtr*> holds
ex g, f1, f2 being Function of NAT ,(the Instructions of SCM ^omega ) st
( g = f . (nt -tree t1,t2) & f1 = f . t1 & f2 = f . t2 & ( for n being Element of NAT holds g . n = ((f1 . n) ^ (f2 . (n + 1))) ^ (Selfwork nt,n) ) ) ) ) ; :: thesis: verum