let S1, S2 be non empty non void unsplit gate`1=arity gate`2isBoolean ManySortedSign ; :: thesis: for A1 being gate`2=den Boolean Circuit of S1
for A2 being gate`2=den Boolean Circuit of S2
for s being State of (A1 +* A2)
for v being Vertex of (S1 +* S2) holds
( ( for s1 being State of A1 st s1 = s | the carrier of S1 & ( v in InnerVertices S1 or ( v in the carrier of S1 & v in InputVertices (S1 +* S2) ) ) holds
(Following s) . v = (Following s1) . v ) & ( for s2 being State of A2 st s2 = s | the carrier of S2 & ( v in InnerVertices S2 or ( v in the carrier of S2 & v in InputVertices (S1 +* S2) ) ) holds
(Following s) . v = (Following s2) . v ) )
let A1 be gate`2=den Boolean Circuit of S1; :: thesis: for A2 being gate`2=den Boolean Circuit of S2
for s being State of (A1 +* A2)
for v being Vertex of (S1 +* S2) holds
( ( for s1 being State of A1 st s1 = s | the carrier of S1 & ( v in InnerVertices S1 or ( v in the carrier of S1 & v in InputVertices (S1 +* S2) ) ) holds
(Following s) . v = (Following s1) . v ) & ( for s2 being State of A2 st s2 = s | the carrier of S2 & ( v in InnerVertices S2 or ( v in the carrier of S2 & v in InputVertices (S1 +* S2) ) ) holds
(Following s) . v = (Following s2) . v ) )
let A2 be gate`2=den Boolean Circuit of S2; :: thesis: for s being State of (A1 +* A2)
for v being Vertex of (S1 +* S2) holds
( ( for s1 being State of A1 st s1 = s | the carrier of S1 & ( v in InnerVertices S1 or ( v in the carrier of S1 & v in InputVertices (S1 +* S2) ) ) holds
(Following s) . v = (Following s1) . v ) & ( for s2 being State of A2 st s2 = s | the carrier of S2 & ( v in InnerVertices S2 or ( v in the carrier of S2 & v in InputVertices (S1 +* S2) ) ) holds
(Following s) . v = (Following s2) . v ) )
A1 tolerates A2 by Th60;
hence for s being State of (A1 +* A2)
for v being Vertex of (S1 +* S2) holds
( ( for s1 being State of A1 st s1 = s | the carrier of S1 & ( v in InnerVertices S1 or ( v in the carrier of S1 & v in InputVertices (S1 +* S2) ) ) holds
(Following s) . v = (Following s1) . v ) & ( for s2 being State of A2 st s2 = s | the carrier of S2 & ( v in InnerVertices S2 or ( v in the carrier of S2 & v in InputVertices (S1 +* S2) ) ) holds
(Following s) . v = (Following s2) . v ) ) by Th31; :: thesis: verum