let X be non empty disjoint_with_NAT set ; :: thesis: for I1, I2 being Element of (FreeUnivAlgNSG (ECIW-signature,X)) holds I1 \; I2 = 2 -tree (I1,I2)
set S = ECIW-signature ;
reconsider s = ECIW-signature as non empty FinSequence of omega ;
set A = FreeUnivAlgNSG (ECIW-signature,X);
let I1, I2 be Element of (FreeUnivAlgNSG (ECIW-signature,X)); :: thesis: I1 \; I2 = 2 -tree (I1,I2)
A1: 2 in dom the charact of (FreeUnivAlgNSG (ECIW-signature,X)) by Def11;
reconsider f = the charact of (FreeUnivAlgNSG (ECIW-signature,X)) . 2 as non empty homogeneous quasi_total 2 -ary PartFunc of ( the carrier of (FreeUnivAlgNSG (ECIW-signature,X)) *), the carrier of (FreeUnivAlgNSG (ECIW-signature,X)) by Def11;
A2: f = FreeOpNSG (2,ECIW-signature,X) by A1, FREEALG:def 11;
A3: 2 in dom ECIW-signature by Th54;
then s /. 2 = ECIW-signature . 2 by PARTFUN1:def 6;
then A4: dom (FreeOpNSG (2,ECIW-signature,X)) = 2 -tuples_on (TS (DTConUA (ECIW-signature,X))) by A3, Th54, FREEALG:def 10;
A5: <*I1,I2*> in 2 -tuples_on (TS (DTConUA (ECIW-signature,X))) by FINSEQ_2:137;
thus I1 \; I2 = f . <*I1,I2*> by A1, SUBSET_1:def 8
.= (Sym (2,ECIW-signature,X)) -tree <*I1,I2*> by A2, A3, A4, A5, FREEALG:def 10
.= 2 -tree (I1,I2) by A3, FREEALG:def 9 ; :: thesis: verum