set AL = (((FreeSort X) #) * the Arity of S) . o;
set AX = ((FreeSort X) * the ResultSort of S) . o;
set D = DTConMSA X;
let f, g be Function of ((((FreeSort X) #) * the Arity of S) . o),(((FreeSort X) * the ResultSort of S) . o); :: thesis: ( ( for p being FinSequence of TS (DTConMSA X) st Sym (o,X) ==> roots p holds
f . p = (Sym (o,X)) -tree p ) & ( for p being FinSequence of TS (DTConMSA X) st Sym (o,X) ==> roots p holds
g . p = (Sym (o,X)) -tree p ) implies f = g )
assume that
A5: for p being FinSequence of TS (DTConMSA X) st Sym (o,X) ==> roots p holds
f . p = (Sym (o,X)) -tree p and
A6: for p being FinSequence of TS (DTConMSA X) st Sym (o,X) ==> roots p holds
g . p = (Sym (o,X)) -tree p ; :: thesis: f = g
A7: for x being set st x in (((FreeSort X) #) * the Arity of S) . o holds
f . x = g . x

proof

let x be set ; :: thesis: ( x in (((FreeSort X) #) * the Arity of S) . o implies f . x = g . x )
assume A8: x in (((FreeSort X) #) * the Arity of S) . o ; :: thesis: f . x = g . x
then reconsider p = x as FinSequence of TS (DTConMSA X) by Th8;
A9: Sym (o,X) ==> roots p by A8, Th10;
then f . p = (Sym (o,X)) -tree p by A5;
hence f . x = g . x by A6, A9; :: thesis: verum

end;

( dom f = (((FreeSort X) #) * the Arity of S) . o & dom g = (((FreeSort X) #) * the Arity of S) . o ) by FUNCT_2:def 1;
hence f = g by A7, FUNCT_1:9; :: thesis: verum