let X be non empty disjoint_with_NAT set ; :: thesis: for p being FinSequence of (FreeUnivAlgNSG ECIW-signature ,X) st 1 -tree p is Element of (FreeUnivAlgNSG ECIW-signature ,X) holds
p = {}
set S = ECIW-signature ;
set G = DTConUA ECIW-signature ,X;
set A = FreeUnivAlgNSG ECIW-signature ,X;
let p be FinSequence of (FreeUnivAlgNSG ECIW-signature ,X); :: thesis: ( 1 -tree p is Element of (FreeUnivAlgNSG ECIW-signature ,X) implies p = {} )
assume 1 -tree p is Element of (FreeUnivAlgNSG ECIW-signature ,X) ; :: thesis: p = {}
then reconsider I = 1 -tree p as Element of (FreeUnivAlgNSG ECIW-signature ,X) ;

per cases ( ex x being Element of X st I = root-tree x or ex n being Nat ex p being FinSequence of (FreeUnivAlgNSG ECIW-signature ,X) st
( n in Seg 4 & I = n -tree p & len p = ECIW-signature . n ) ) by Th56;

suppose ex x being Element of X st I = root-tree x ; :: thesis: p = {}

then consider x being Element of X such that
A1: 1 -tree p = root-tree x ;
1 -tree p = x -tree (<*> (TS (DTConUA ECIW-signature ,X))) by A1, TREES_4:20;
hence p = {} by TREES_4:15; :: thesis: verum

end;

suppose ex n being Nat ex p being FinSequence of (FreeUnivAlgNSG ECIW-signature ,X) st
( n in Seg 4 & I = n -tree p & len p = ECIW-signature . n ) ; :: thesis: p = {}

then consider n being Nat, q being FinSequence of (FreeUnivAlgNSG ECIW-signature ,X) such that
A2: ( n in Seg 4 & I = n -tree q & len q = ECIW-signature . n ) ;
( n = 1 & q = p ) by A2, TREES_4:15;
hence p = {} by A2, Th54; :: thesis: verum

end;

end;