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 ternary 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 12;
A3: 3 in dom ECIW-signature by Th54;
then s /. 3 = ECIW-signature . 3 by PARTFUN1:def 8;
then A4: dom (FreeOpNSG 3,ECIW-signature ,X) = 3 -tuples_on (TS (DTConUA ECIW-signature ,X)) by A3, Th54, FREEALG:def 11;
A5: <*C,I1,I2*> in 3 -tuples_on (TS (DTConUA ECIW-signature ,X)) by FINSEQ_2:159;
thus if-then-else C,I1,I2 = f . <*C,I1,I2*> by A1, FUNCT_7:def 1
.= (Sym 3,ECIW-signature ,X) -tree <*C,I1,I2*> by A2, A3, A4, A5, FREEALG:def 11
.= 3 -tree <*C,I1,I2*> by A3, FREEALG:def 10 ; :: thesis: verum