let G1, G2 be non empty non void Circuit-like ManySortedSign ; for f, g being Function
for C1 being non-empty Circuit of G1
for C2 being non-empty Circuit of G2 st f,g form_embedding_of C1,C2 holds
( dom f = the carrier of G1 & rng f c= the carrier of G2 & dom g = the carrier' of G1 & rng g c= the carrier' of G2 )
let f, g be Function; for C1 being non-empty Circuit of G1
for C2 being non-empty Circuit of G2 st f,g form_embedding_of C1,C2 holds
( dom f = the carrier of G1 & rng f c= the carrier of G2 & dom g = the carrier' of G1 & rng g c= the carrier' of G2 )
let C1 be non-empty Circuit of G1; for C2 being non-empty Circuit of G2 st f,g form_embedding_of C1,C2 holds
( dom f = the carrier of G1 & rng f c= the carrier of G2 & dom g = the carrier' of G1 & rng g c= the carrier' of G2 )
let C2 be non-empty Circuit of G2; ( f,g form_embedding_of C1,C2 implies ( dom f = the carrier of G1 & rng f c= the carrier of G2 & dom g = the carrier' of G1 & rng g c= the carrier' of G2 ) )
assume that
f is one-to-one
and
g is one-to-one
and
A1:
dom f = the carrier of G1
and
A2:
dom g = the carrier' of G1
and
A3:
rng f c= the carrier of G2
and
A4:
rng g c= the carrier' of G2
; PUA2MSS1:def 12,CIRCTRM1:def 12 ( not the ResultSort of G1 * f = g * the ResultSort of G2 or ex b1 being set ex b2 being set st
( b1 in the carrier' of G1 & b2 = the Arity of G1 . b1 & not b2 * f = the Arity of G2 . (g . b1) ) or not the Sorts of C1 = the Sorts of C2 * f or not the Charact of C1 = the Charact of C2 * g or ( dom f = the carrier of G1 & rng f c= the carrier of G2 & dom g = the carrier' of G1 & rng g c= the carrier' of G2 ) )
thus
( not the ResultSort of G1 * f = g * the ResultSort of G2 or ex b1 being set ex b2 being set st
( b1 in the carrier' of G1 & b2 = the Arity of G1 . b1 & not b2 * f = the Arity of G2 . (g . b1) ) or not the Sorts of C1 = the Sorts of C2 * f or not the Charact of C1 = the Charact of C2 * g or ( dom f = the carrier of G1 & rng f c= the carrier of G2 & dom g = the carrier' of G1 & rng g c= the carrier' of G2 ) )
by A1, A2, A3, A4; verum