let S1, S2 be non empty Signature; :: thesis: S1 +* S2 is Extension of S2
set S = S1 +* S2;
set f1 = id the carrier of S2;
set g1 = id the carrier' of S2;
thus ( dom (id the carrier of S2) = the carrier of S2 & dom (id the carrier' of S2) = the carrier' of S2 ) by RELAT_1:71; :: according to PUA2MSS1:def 13,INSTALG1:def 2,ALGSPEC1:def 5 :: thesis: ( rng (id the carrier of S2) c= the carrier of (S1 +* S2) & rng (id the carrier' of S2) c= the carrier' of (S1 +* S2) & the ResultSort of S2 * (id the carrier of S2) = (id the carrier' of S2) * the ResultSort of (S1 +* S2) & ( for b₁ being set
for b₂ being set holds
( not b₁ in the carrier' of S2 or not b₂ = the Arity of S2 . b₁ or b₂ * (id the carrier of S2) = the Arity of (S1 +* S2) . ((id the carrier' of S2) . b₁) ) ) )
A1: ( rng (id the carrier of S2) = the carrier of S2 & rng (id the carrier' of S2) = the carrier' of S2 ) by RELAT_1:71;
( the carrier of (S1 +* S2) = the carrier of S1 \/ the carrier of S2 & the carrier' of (S1 +* S2) = the carrier' of S1 \/ the carrier' of S2 ) by CIRCCOMB:def 2;
hence ( rng (id the carrier of S2) c= the carrier of (S1 +* S2) & rng (id the carrier' of S2) c= the carrier' of (S1 +* S2) ) by A1, XBOOLE_1:7; :: thesis: ( the ResultSort of S2 * (id the carrier of S2) = (id the carrier' of S2) * the ResultSort of (S1 +* S2) & ( for b₁ being set
for b₂ being set holds
( not b₁ in the carrier' of S2 or not b₂ = the Arity of S2 . b₁ or b₂ * (id the carrier of S2) = the Arity of (S1 +* S2) . ((id the carrier' of S2) . b₁) ) ) )
A2: ( dom the ResultSort of S2 = the carrier' of S2 & rng the ResultSort of S2 c= the carrier of S2 ) by FUNCT_2:def 1;
the ResultSort of (S1 +* S2) = the ResultSort of S1 +* the ResultSort of S2 by CIRCCOMB:def 2;
then the ResultSort of S2 = the ResultSort of (S1 +* S2) | the carrier' of S2 by A2, FUNCT_4:26, GRFUNC_1:64;
then the ResultSort of S2 = the ResultSort of (S1 +* S2) * (id the carrier' of S2) by RELAT_1:94;
hence (id the carrier of S2) * the ResultSort of S2 = the ResultSort of (S1 +* S2) * (id the carrier' of S2) by A2, RELAT_1:79; :: thesis: for b₁ being set
for b₂ being set holds
( not b₁ in the carrier' of S2 or not b₂ = the Arity of S2 . b₁ or b₂ * (id the carrier of S2) = the Arity of (S1 +* S2) . ((id the carrier' of S2) . b₁) )
let o be set ; :: thesis: for b₁ being set holds
( not o in the carrier' of S2 or not b₁ = the Arity of S2 . o or b₁ * (id the carrier of S2) = the Arity of (S1 +* S2) . ((id the carrier' of S2) . o) )
let p be Function; :: thesis: ( not o in the carrier' of S2 or not p = the Arity of S2 . o or p * (id the carrier of S2) = the Arity of (S1 +* S2) . ((id the carrier' of S2) . o) )
assume A3: ( o in the carrier' of S2 & p = the Arity of S2 . o ) ; :: thesis: p * (id the carrier of S2) = the Arity of (S1 +* S2) . ((id the carrier' of S2) . o)
A4: ( dom the Arity of S2 = the carrier' of S2 & rng the Arity of S2 c= the carrier of S2 * ) by FUNCT_2:def 1;
then p in rng the Arity of S2 by A3, FUNCT_1:def 5;
then p is FinSequence of the carrier of S2 by FINSEQ_1:def 11;
then rng p c= the carrier of S2 by FINSEQ_1:def 4;
hence (id the carrier of S2) * p = p by RELAT_1:79
.= (the Arity of S1 +* the Arity of S2) . o by A3, A4, FUNCT_4:14
.= the Arity of (S1 +* S2) . o by CIRCCOMB:def 2
.= the Arity of (S1 +* S2) . ((id the carrier' of S2) . o) by A3, FUNCT_1:35 ;
:: thesis: verum