let X be non empty disjoint_with_NAT set ; :: thesis: for C, I1, I2 being Element of (FreeUnivAlgNSG (ECIW-signature,X)) holds if-then-else (C,I1,I2) = 3 -tree <*C,I1,I2*>
set S = ECIW-signature ;
reconsider s = ECIW-signature as non empty FinSequence of omega ;
set A = FreeUnivAlgNSG (ECIW-signature,X);
let C, I1, I2 be Element of (FreeUnivAlgNSG (ECIW-signature,X)); :: thesis: if-then-else (C,I1,I2) = 3 -tree <*C,I1,I2*>
A1: 3 in dom the charact of (FreeUnivAlgNSG (ECIW-signature,X)) by Def12;
reconsider f = the charact of (FreeUnivAlgNSG (ECIW-signature,X)) . 3 as non empty homogeneous quasi_total 3 -ary PartFunc of ( the carrier of (FreeUnivAlgNSG (ECIW-signature,X)) *), the carrier of (FreeUnivAlgNSG (ECIW-signature,X)) by Def12;
A2: f = FreeOpNSG (3,ECIW-signature,X) by A1, FREEALG:def 11;
A3: 3 in dom ECIW-signature by Th54;
then s /. 3 = ECIW-signature . 3 by PARTFUN1:def 6;
then A4: dom (FreeOpNSG (3,ECIW-signature,X)) = 3 -tuples_on (TS (DTConUA (ECIW-signature,X))) by A3, Th54, FREEALG:def 10;
A5: <*C,I1,I2*> in 3 -tuples_on (TS (DTConUA (ECIW-signature,X))) by FINSEQ_2:139;
thus if-then-else (C,I1,I2) = f . <*C,I1,I2*> by A1, SUBSET_1:def 8
.= (Sym (3,ECIW-signature,X)) -tree <*C,I1,I2*> by A2, A3, A4, A5, FREEALG:def 10
.= 3 -tree <*C,I1,I2*> by A3, FREEALG:def 9 ; :: thesis: verum