consider f being Function of (TS SCM-AE),(Funcs (NAT,( the InstructionsF of SCM ^omega))) such that

A1: ( ( for t being Terminal of SCM-AE ex g being sequence of ( the InstructionsF of SCM ^omega) st

( g = f . (root-tree t) & ( for n being Element of NAT holds g . n = H_{3}(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 sequence of ( the InstructionsF 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 = H_{2}(nt,f1,f2,n) ) ) ) )
from BINTREE1:sch 5();

take f ; :: thesis: ( ( for t being Terminal of SCM-AE ex g being sequence of ( the InstructionsF of SCM ^omega) st

( g = f . (root-tree t) & ( for n being 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 sequence of ( the InstructionsF of SCM ^omega) st

( g = f . (nt -tree (t1,t2)) & f1 = f . t1 & f2 = f . t2 & ( for n being Nat holds g . n = ((f1 . (In (n,NAT))) ^ (f2 . (In ((n + 1),NAT)))) ^ (Selfwork (nt,n)) ) ) ) )

( ( for t being Terminal of SCM-AE ex g being sequence of ( the InstructionsF of SCM ^omega) st

( g = f . (root-tree t) & ( for n being 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 sequence of ( the InstructionsF of SCM ^omega) st

( g = f . (nt -tree (t1,t2)) & f1 = f . t1 & f2 = f . t2 & ( for n being Nat holds g . n = ((f1 . (In (n,NAT))) ^ (f2 . (In ((n + 1),NAT)))) ^ (Selfwork (nt,n)) ) ) ) )

( g = f . (root-tree t) & ( for n being 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 sequence of ( the InstructionsF of SCM ^omega) st

( g = f . (nt -tree (t1,t2)) & f1 = f . t1 & f2 = f . t2 & ( for n being Nat holds g . n = ((f1 . (In (n,NAT))) ^ (f2 . (In ((n + 1),NAT)))) ^ (Selfwork (nt,n)) ) ) ) ) ; :: thesis: verum

A1: ( ( for t being Terminal of SCM-AE ex g being sequence of ( the InstructionsF of SCM ^omega) st

( g = f . (root-tree t) & ( for n being Element of NAT holds g . n = H

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 sequence of ( the InstructionsF 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 = H

take f ; :: thesis: ( ( for t being Terminal of SCM-AE ex g being sequence of ( the InstructionsF of SCM ^omega) st

( g = f . (root-tree t) & ( for n being 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 sequence of ( the InstructionsF of SCM ^omega) st

( g = f . (nt -tree (t1,t2)) & f1 = f . t1 & f2 = f . t2 & ( for n being Nat holds g . n = ((f1 . (In (n,NAT))) ^ (f2 . (In ((n + 1),NAT)))) ^ (Selfwork (nt,n)) ) ) ) )

( ( for t being Terminal of SCM-AE ex g being sequence of ( the InstructionsF of SCM ^omega) st

( g = f . (root-tree t) & ( for n being 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 sequence of ( the InstructionsF of SCM ^omega) st

( g = f . (nt -tree (t1,t2)) & f1 = f . t1 & f2 = f . t2 & ( for n being Nat holds g . n = ((f1 . (In (n,NAT))) ^ (f2 . (In ((n + 1),NAT)))) ^ (Selfwork (nt,n)) ) ) ) )

proof

hence
( ( for t being Terminal of SCM-AE ex g being sequence of ( the InstructionsF of SCM ^omega) st
thus
for t being Terminal of SCM-AE ex g being sequence of ( the InstructionsF of SCM ^omega) st

( g = f . (root-tree t) & ( for n being 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 sequence of ( the InstructionsF of SCM ^omega) st

( g = f . (nt -tree (t1,t2)) & f1 = f . t1 & f2 = f . t2 & ( for n being Nat holds g . n = ((f1 . (In (n,NAT))) ^ (f2 . (In ((n + 1),NAT)))) ^ (Selfwork (nt,n)) ) )

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 sequence of ( the InstructionsF of SCM ^omega) st

( g = f . (nt -tree (t1,t2)) & f1 = f . t1 & f2 = f . t2 & ( for n being Nat holds g . n = ((f1 . (In (n,NAT))) ^ (f2 . (In ((n + 1),NAT)))) ^ (Selfwork (nt,n)) ) ) :: thesis: verum

end;( g = f . (root-tree t) & ( for n being 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 sequence of ( the InstructionsF of SCM ^omega) st

( g = f . (nt -tree (t1,t2)) & f1 = f . t1 & f2 = f . t2 & ( for n being Nat holds g . n = ((f1 . (In (n,NAT))) ^ (f2 . (In ((n + 1),NAT)))) ^ (Selfwork (nt,n)) ) )

proof

thus
for nt being NonTerminal of SCM-AE
let t be Terminal of SCM-AE; :: thesis: ex g being sequence of ( the InstructionsF of SCM ^omega) st

( g = f . (root-tree t) & ( for n being Nat holds g . n = <%((dl. n) := (@ t))%> ) )

consider g being sequence of ( the InstructionsF of SCM ^omega) such that

A2: g = f . (root-tree t) and

A3: for n being Element of NAT holds g . n = H_{3}(t,n)
by A1;

take g ; :: thesis: ( g = f . (root-tree t) & ( for n being Nat holds g . n = <%((dl. n) := (@ t))%> ) )

thus g = f . (root-tree t) by A2; :: thesis: for n being Nat holds g . n = <%((dl. n) := (@ t))%>

let n be Nat; :: thesis: g . n = <%((dl. n) := (@ t))%>

n in NAT by ORDINAL1:def 12;

then g . n = H_{3}(t,n)
by A3;

hence g . n = <%((dl. n) := (@ t))%> ; :: thesis: verum

end;( g = f . (root-tree t) & ( for n being Nat holds g . n = <%((dl. n) := (@ t))%> ) )

consider g being sequence of ( the InstructionsF of SCM ^omega) such that

A2: g = f . (root-tree t) and

A3: for n being Element of NAT holds g . n = H

take g ; :: thesis: ( g = f . (root-tree t) & ( for n being Nat holds g . n = <%((dl. n) := (@ t))%> ) )

thus g = f . (root-tree t) by A2; :: thesis: for n being Nat holds g . n = <%((dl. n) := (@ t))%>

let n be Nat; :: thesis: g . n = <%((dl. n) := (@ t))%>

n in NAT by ORDINAL1:def 12;

then g . n = H

hence g . n = <%((dl. n) := (@ t))%> ; :: thesis: verum

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 sequence of ( the InstructionsF of SCM ^omega) st

( g = f . (nt -tree (t1,t2)) & f1 = f . t1 & f2 = f . t2 & ( for n being Nat holds g . n = ((f1 . (In (n,NAT))) ^ (f2 . (In ((n + 1),NAT)))) ^ (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 sequence of ( the InstructionsF of SCM ^omega) st

( g = f . (nt -tree (t1,t2)) & f1 = f . t1 & f2 = f . t2 & ( for n being Nat holds g . n = ((f1 . (In (n,NAT))) ^ (f2 . (In ((n + 1),NAT)))) ^ (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 sequence of ( the InstructionsF of SCM ^omega) st

( g = f . (nt -tree (t1,t2)) & f1 = f . t1 & f2 = f . t2 & ( for n being Nat holds g . n = ((f1 . (In (n,NAT))) ^ (f2 . (In ((n + 1),NAT)))) ^ (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 sequence of ( the InstructionsF of SCM ^omega) st

( g = f . (nt -tree (t1,t2)) & f1 = f . t1 & f2 = f . t2 & ( for n being Nat holds g . n = ((f1 . (In (n,NAT))) ^ (f2 . (In ((n + 1),NAT)))) ^ (Selfwork (nt,n)) ) ) )

assume A4: ( rtl = root-label t1 & rtr = root-label t2 & nt ==> <*rtl,rtr*> ) ; :: thesis: ex g, f1, f2 being sequence of ( the InstructionsF of SCM ^omega) st

( g = f . (nt -tree (t1,t2)) & f1 = f . t1 & f2 = f . t2 & ( for n being Nat holds g . n = ((f1 . (In (n,NAT))) ^ (f2 . (In ((n + 1),NAT)))) ^ (Selfwork (nt,n)) ) )

consider g, f1, f2 being sequence of ( the InstructionsF of SCM ^omega) such that

A5: ( g = f . (nt -tree (t1,t2)) & f1 = f . t1 & f2 = f . t2 ) and

A6: for n being Element of NAT holds g . n = H_{2}(nt,f1,f2,n)
by A4, A1;

take g ; :: thesis: ex f1, f2 being sequence of ( the InstructionsF of SCM ^omega) st

( g = f . (nt -tree (t1,t2)) & f1 = f . t1 & f2 = f . t2 & ( for n being Nat holds g . n = ((f1 . (In (n,NAT))) ^ (f2 . (In ((n + 1),NAT)))) ^ (Selfwork (nt,n)) ) )

take f1 ; :: thesis: ex f2 being sequence of ( the InstructionsF of SCM ^omega) st

( g = f . (nt -tree (t1,t2)) & f1 = f . t1 & f2 = f . t2 & ( for n being Nat holds g . n = ((f1 . (In (n,NAT))) ^ (f2 . (In ((n + 1),NAT)))) ^ (Selfwork (nt,n)) ) )

take f2 ; :: thesis: ( g = f . (nt -tree (t1,t2)) & f1 = f . t1 & f2 = f . t2 & ( for n being Nat holds g . n = ((f1 . (In (n,NAT))) ^ (f2 . (In ((n + 1),NAT)))) ^ (Selfwork (nt,n)) ) )

thus ( g = f . (nt -tree (t1,t2)) & f1 = f . t1 & f2 = f . t2 ) by A5; :: thesis: for n being Nat holds g . n = ((f1 . (In (n,NAT))) ^ (f2 . (In ((n + 1),NAT)))) ^ (Selfwork (nt,n))

let n be Nat; :: thesis: g . n = ((f1 . (In (n,NAT))) ^ (f2 . (In ((n + 1),NAT)))) ^ (Selfwork (nt,n))

g . n = H_{2}(nt,f1,f2,n)
by A6;

hence g . n = ((f1 . (In (n,NAT))) ^ (f2 . (In ((n + 1),NAT)))) ^ (Selfwork (nt,n)) ; :: thesis: verum

end;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 sequence of ( the InstructionsF of SCM ^omega) st

( g = f . (nt -tree (t1,t2)) & f1 = f . t1 & f2 = f . t2 & ( for n being Nat holds g . n = ((f1 . (In (n,NAT))) ^ (f2 . (In ((n + 1),NAT)))) ^ (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 sequence of ( the InstructionsF of SCM ^omega) st

( g = f . (nt -tree (t1,t2)) & f1 = f . t1 & f2 = f . t2 & ( for n being Nat holds g . n = ((f1 . (In (n,NAT))) ^ (f2 . (In ((n + 1),NAT)))) ^ (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 sequence of ( the InstructionsF of SCM ^omega) st

( g = f . (nt -tree (t1,t2)) & f1 = f . t1 & f2 = f . t2 & ( for n being Nat holds g . n = ((f1 . (In (n,NAT))) ^ (f2 . (In ((n + 1),NAT)))) ^ (Selfwork (nt,n)) ) ) )

assume A4: ( rtl = root-label t1 & rtr = root-label t2 & nt ==> <*rtl,rtr*> ) ; :: thesis: ex g, f1, f2 being sequence of ( the InstructionsF of SCM ^omega) st

( g = f . (nt -tree (t1,t2)) & f1 = f . t1 & f2 = f . t2 & ( for n being Nat holds g . n = ((f1 . (In (n,NAT))) ^ (f2 . (In ((n + 1),NAT)))) ^ (Selfwork (nt,n)) ) )

consider g, f1, f2 being sequence of ( the InstructionsF of SCM ^omega) such that

A5: ( g = f . (nt -tree (t1,t2)) & f1 = f . t1 & f2 = f . t2 ) and

A6: for n being Element of NAT holds g . n = H

take g ; :: thesis: ex f1, f2 being sequence of ( the InstructionsF of SCM ^omega) st

( g = f . (nt -tree (t1,t2)) & f1 = f . t1 & f2 = f . t2 & ( for n being Nat holds g . n = ((f1 . (In (n,NAT))) ^ (f2 . (In ((n + 1),NAT)))) ^ (Selfwork (nt,n)) ) )

take f1 ; :: thesis: ex f2 being sequence of ( the InstructionsF of SCM ^omega) st

( g = f . (nt -tree (t1,t2)) & f1 = f . t1 & f2 = f . t2 & ( for n being Nat holds g . n = ((f1 . (In (n,NAT))) ^ (f2 . (In ((n + 1),NAT)))) ^ (Selfwork (nt,n)) ) )

take f2 ; :: thesis: ( g = f . (nt -tree (t1,t2)) & f1 = f . t1 & f2 = f . t2 & ( for n being Nat holds g . n = ((f1 . (In (n,NAT))) ^ (f2 . (In ((n + 1),NAT)))) ^ (Selfwork (nt,n)) ) )

thus ( g = f . (nt -tree (t1,t2)) & f1 = f . t1 & f2 = f . t2 ) by A5; :: thesis: for n being Nat holds g . n = ((f1 . (In (n,NAT))) ^ (f2 . (In ((n + 1),NAT)))) ^ (Selfwork (nt,n))

let n be Nat; :: thesis: g . n = ((f1 . (In (n,NAT))) ^ (f2 . (In ((n + 1),NAT)))) ^ (Selfwork (nt,n))

g . n = H

hence g . n = ((f1 . (In (n,NAT))) ^ (f2 . (In ((n + 1),NAT)))) ^ (Selfwork (nt,n)) ; :: thesis: verum

( g = f . (root-tree t) & ( for n being 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 sequence of ( the InstructionsF of SCM ^omega) st

( g = f . (nt -tree (t1,t2)) & f1 = f . t1 & f2 = f . t2 & ( for n being Nat holds g . n = ((f1 . (In (n,NAT))) ^ (f2 . (In ((n + 1),NAT)))) ^ (Selfwork (nt,n)) ) ) ) ) ; :: thesis: verum