let f1, f2 be Function of (),(Funcs (NAT,())); :: thesis: ( ( for t being Terminal of SCM-AE ex g being sequence of () st
( g = f1 . () & ( 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 () st
( g = f1 . (nt -tree (t1,t2)) & f1 = f1 . t1 & f2 = f1 . 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 () st
( g = f2 . () & ( 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 () st
( g = f2 . (nt -tree (t1,t2)) & f1 = f2 . t1 & f2 = f2 . t2 & ( for n being Nat holds g . n = ((f1 . (In (n,NAT))) ^ (f2 . (In ((n + 1),NAT)))) ^ (Selfwork (nt,n)) ) ) ) implies f1 = f2 )

assume that
A7: ( ( for t being Terminal of SCM-AE ex g being sequence of () st
( g = f1 . () & ( 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, g1, g2 being sequence of () st
( g = f1 . (nt -tree (t1,t2)) & g1 = f1 . t1 & g2 = f1 . t2 & ( for n being Nat holds g . n = ((g1 . (In (n,NAT))) ^ (g2 . (In ((n + 1),NAT)))) ^ (Selfwork (nt,n)) ) ) ) ) and
A8: ( ( for t being Terminal of SCM-AE ex g being sequence of () st
( g = f2 . () & ( 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, g1, g2 being sequence of () st
( g = f2 . (nt -tree (t1,t2)) & g1 = f2 . t1 & g2 = f2 . t2 & ( for n being Nat holds g . n = ((g1 . (In (n,NAT))) ^ (g2 . (In ((n + 1),NAT)))) ^ (Selfwork (nt,n)) ) ) ) ) ; :: thesis: f1 = f2
A9: ( ( for t being Terminal of SCM-AE ex g being sequence of () st
( g = f1 . () & ( 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, g1, g2 being sequence of () st
( g = f1 . (nt -tree (t1,t2)) & g1 = f1 . t1 & g2 = f1 . t2 & ( for n being Element of NAT holds g . n = H2(nt,g1,g2,n) ) ) ) )
proof
thus for t being Terminal of SCM-AE ex g being sequence of () st
( g = f1 . () & ( for n being Element of NAT holds g . n = H3(t,n) ) ) :: 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, g1, g2 being sequence of () st
( g = f1 . (nt -tree (t1,t2)) & g1 = f1 . t1 & g2 = f1 . t2 & ( for n being Element of NAT holds g . n = H2(nt,g1,g2,n) ) )
proof
let t be Terminal of SCM-AE; :: thesis: ex g being sequence of () st
( g = f1 . () & ( for n being Element of NAT holds g . n = H3(t,n) ) )

consider g being sequence of () such that
A10: ( g = f1 . () & ( for n being Nat holds g . n = <%((dl. n) := (@ t))%> ) ) by A7;
take g ; :: thesis: ( g = f1 . () & ( for n being Element of NAT holds g . n = H3(t,n) ) )
thus ( g = f1 . () & ( for n being Element of NAT holds g . n = H3(t,n) ) ) by A10; :: 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, g1, g2 being sequence of () st
( g = f1 . (nt -tree (t1,t2)) & g1 = f1 . t1 & g2 = f1 . t2 & ( for n being Element of NAT holds g . n = H2(nt,g1,g2,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, g1, g2 being sequence of () st
( g = f1 . (nt -tree (t1,t2)) & g1 = f1 . t1 & g2 = f1 . t2 & ( for n being Element of NAT holds g . n = H2(nt,g1,g2,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, g1, g2 being sequence of () st
( g = f1 . (nt -tree (t1,t2)) & g1 = f1 . t1 & g2 = f1 . t2 & ( for n being Element of NAT holds g . n = H2(nt,g1,g2,n) ) )

let rtl, rtr be Symbol of SCM-AE; :: thesis: ( rtl = root-label t1 & rtr = root-label t2 & nt ==> <*rtl,rtr*> implies ex g, g1, g2 being sequence of () st
( g = f1 . (nt -tree (t1,t2)) & g1 = f1 . t1 & g2 = f1 . t2 & ( for n being Element of NAT holds g . n = H2(nt,g1,g2,n) ) ) )

assume A11: ( rtl = root-label t1 & rtr = root-label t2 & nt ==> <*rtl,rtr*> ) ; :: thesis: ex g, g1, g2 being sequence of () st
( g = f1 . (nt -tree (t1,t2)) & g1 = f1 . t1 & g2 = f1 . t2 & ( for n being Element of NAT holds g . n = H2(nt,g1,g2,n) ) )

consider g, g1, g2 being sequence of () such that
A12: ( g = f1 . (nt -tree (t1,t2)) & g1 = f1 . t1 & g2 = f1 . t2 & ( for n being Nat holds g . n = ((g1 . (In (n,NAT))) ^ (g2 . (In ((n + 1),NAT)))) ^ (Selfwork (nt,n)) ) ) by ;
take g ; :: thesis: ex g1, g2 being sequence of () st
( g = f1 . (nt -tree (t1,t2)) & g1 = f1 . t1 & g2 = f1 . t2 & ( for n being Element of NAT holds g . n = H2(nt,g1,g2,n) ) )

take g1 ; :: thesis: ex g2 being sequence of () st
( g = f1 . (nt -tree (t1,t2)) & g1 = f1 . t1 & g2 = f1 . t2 & ( for n being Element of NAT holds g . n = H2(nt,g1,g2,n) ) )

take g2 ; :: thesis: ( g = f1 . (nt -tree (t1,t2)) & g1 = f1 . t1 & g2 = f1 . t2 & ( for n being Element of NAT holds g . n = H2(nt,g1,g2,n) ) )
thus ( g = f1 . (nt -tree (t1,t2)) & g1 = f1 . t1 & g2 = f1 . t2 & ( for n being Element of NAT holds g . n = H2(nt,g1,g2,n) ) ) by A12; :: thesis: verum
end;
end;
A13: ( ( for t being Terminal of SCM-AE ex g being sequence of () st
( g = f2 . () & ( 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, g1, g2 being sequence of () st
( g = f2 . (nt -tree (t1,t2)) & g1 = f2 . t1 & g2 = f2 . t2 & ( for n being Element of NAT holds g . n = H2(nt,g1,g2,n) ) ) ) )
proof
thus for t being Terminal of SCM-AE ex g being sequence of () st
( g = f2 . () & ( for n being Element of NAT holds g . n = H3(t,n) ) ) :: 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, g1, g2 being sequence of () st
( g = f2 . (nt -tree (t1,t2)) & g1 = f2 . t1 & g2 = f2 . t2 & ( for n being Element of NAT holds g . n = H2(nt,g1,g2,n) ) )
proof
let t be Terminal of SCM-AE; :: thesis: ex g being sequence of () st
( g = f2 . () & ( for n being Element of NAT holds g . n = H3(t,n) ) )

consider g being sequence of () such that
A14: ( g = f2 . () & ( for n being Nat holds g . n = <%((dl. n) := (@ t))%> ) ) by A8;
take g ; :: thesis: ( g = f2 . () & ( for n being Element of NAT holds g . n = H3(t,n) ) )
thus ( g = f2 . () & ( for n being Element of NAT holds g . n = H3(t,n) ) ) by A14; :: 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, g1, g2 being sequence of () st
( g = f2 . (nt -tree (t1,t2)) & g1 = f2 . t1 & g2 = f2 . t2 & ( for n being Element of NAT holds g . n = H2(nt,g1,g2,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, g1, g2 being sequence of () st
( g = f2 . (nt -tree (t1,t2)) & g1 = f2 . t1 & g2 = f2 . t2 & ( for n being Element of NAT holds g . n = H2(nt,g1,g2,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, g1, g2 being sequence of () st
( g = f2 . (nt -tree (t1,t2)) & g1 = f2 . t1 & g2 = f2 . t2 & ( for n being Element of NAT holds g . n = H2(nt,g1,g2,n) ) )

let rtl, rtr be Symbol of SCM-AE; :: thesis: ( rtl = root-label t1 & rtr = root-label t2 & nt ==> <*rtl,rtr*> implies ex g, g1, g2 being sequence of () st
( g = f2 . (nt -tree (t1,t2)) & g1 = f2 . t1 & g2 = f2 . t2 & ( for n being Element of NAT holds g . n = H2(nt,g1,g2,n) ) ) )

assume A15: ( rtl = root-label t1 & rtr = root-label t2 & nt ==> <*rtl,rtr*> ) ; :: thesis: ex g, g1, g2 being sequence of () st
( g = f2 . (nt -tree (t1,t2)) & g1 = f2 . t1 & g2 = f2 . t2 & ( for n being Element of NAT holds g . n = H2(nt,g1,g2,n) ) )

consider g, g1, g2 being sequence of () such that
A16: ( g = f2 . (nt -tree (t1,t2)) & g1 = f2 . t1 & g2 = f2 . t2 & ( for n being Nat holds g . n = ((g1 . (In (n,NAT))) ^ (g2 . (In ((n + 1),NAT)))) ^ (Selfwork (nt,n)) ) ) by ;
take g ; :: thesis: ex g1, g2 being sequence of () st
( g = f2 . (nt -tree (t1,t2)) & g1 = f2 . t1 & g2 = f2 . t2 & ( for n being Element of NAT holds g . n = H2(nt,g1,g2,n) ) )

take g1 ; :: thesis: ex g2 being sequence of () st
( g = f2 . (nt -tree (t1,t2)) & g1 = f2 . t1 & g2 = f2 . t2 & ( for n being Element of NAT holds g . n = H2(nt,g1,g2,n) ) )

take g2 ; :: thesis: ( g = f2 . (nt -tree (t1,t2)) & g1 = f2 . t1 & g2 = f2 . t2 & ( for n being Element of NAT holds g . n = H2(nt,g1,g2,n) ) )
thus ( g = f2 . (nt -tree (t1,t2)) & g1 = f2 . t1 & g2 = f2 . t2 & ( for n being Element of NAT holds g . n = H2(nt,g1,g2,n) ) ) by A16; :: thesis: verum
end;
end;
thus f1 = f2 from :: thesis: verum