let x1, x2, x3, x5, x6, x7 be non pair set ; :: thesis: for x4 being set st x4 <> [<*(GFA0AdderOutput (x1,x2,x3)),(GFA0AdderOutput (x5,x6,x7))*>,xor2] & x4 <> [<*(GFA0AdderOutput (x1,x2,x3)),(GFA0AdderOutput (x5,x6,x7))*>,and2] & not x4 in InnerVertices (STC0IIStr (x1,x2,x3,x5,x6,x7)) holds
( x1 in InputVertices (STC0IStr (x1,x2,x3,x4,x5,x6,x7)) & x2 in InputVertices (STC0IStr (x1,x2,x3,x4,x5,x6,x7)) & x3 in InputVertices (STC0IStr (x1,x2,x3,x4,x5,x6,x7)) & x4 in InputVertices (STC0IStr (x1,x2,x3,x4,x5,x6,x7)) & x5 in InputVertices (STC0IStr (x1,x2,x3,x4,x5,x6,x7)) & x6 in InputVertices (STC0IStr (x1,x2,x3,x4,x5,x6,x7)) & x7 in InputVertices (STC0IStr (x1,x2,x3,x4,x5,x6,x7)) )
let x4 be set ; :: thesis: ( x4 <> [<*(GFA0AdderOutput (x1,x2,x3)),(GFA0AdderOutput (x5,x6,x7))*>,xor2] & x4 <> [<*(GFA0AdderOutput (x1,x2,x3)),(GFA0AdderOutput (x5,x6,x7))*>,and2] & not x4 in InnerVertices (STC0IIStr (x1,x2,x3,x5,x6,x7)) implies ( x1 in InputVertices (STC0IStr (x1,x2,x3,x4,x5,x6,x7)) & x2 in InputVertices (STC0IStr (x1,x2,x3,x4,x5,x6,x7)) & x3 in InputVertices (STC0IStr (x1,x2,x3,x4,x5,x6,x7)) & x4 in InputVertices (STC0IStr (x1,x2,x3,x4,x5,x6,x7)) & x5 in InputVertices (STC0IStr (x1,x2,x3,x4,x5,x6,x7)) & x6 in InputVertices (STC0IStr (x1,x2,x3,x4,x5,x6,x7)) & x7 in InputVertices (STC0IStr (x1,x2,x3,x4,x5,x6,x7)) ) )
set S = STC0IStr (x1,x2,x3,x4,x5,x6,x7);
set S1 = STC0IIStr (x1,x2,x3,x5,x6,x7);
set A1 = GFA0AdderOutput (x1,x2,x3);
set A2 = GFA0AdderOutput (x5,x6,x7);
set A1A20 = [<*(GFA0AdderOutput (x1,x2,x3)),(GFA0AdderOutput (x5,x6,x7))*>,xor2];
set A1A2 = [<*(GFA0AdderOutput (x1,x2,x3)),(GFA0AdderOutput (x5,x6,x7))*>,and2];
assume ( x4 <> [<*(GFA0AdderOutput (x1,x2,x3)),(GFA0AdderOutput (x5,x6,x7))*>,xor2] & x4 <> [<*(GFA0AdderOutput (x1,x2,x3)),(GFA0AdderOutput (x5,x6,x7))*>,and2] & not x4 in InnerVertices (STC0IIStr (x1,x2,x3,x5,x6,x7)) ) ; :: thesis: ( x1 in InputVertices (STC0IStr (x1,x2,x3,x4,x5,x6,x7)) & x2 in InputVertices (STC0IStr (x1,x2,x3,x4,x5,x6,x7)) & x3 in InputVertices (STC0IStr (x1,x2,x3,x4,x5,x6,x7)) & x4 in InputVertices (STC0IStr (x1,x2,x3,x4,x5,x6,x7)) & x5 in InputVertices (STC0IStr (x1,x2,x3,x4,x5,x6,x7)) & x6 in InputVertices (STC0IStr (x1,x2,x3,x4,x5,x6,x7)) & x7 in InputVertices (STC0IStr (x1,x2,x3,x4,x5,x6,x7)) )
then InputVertices (STC0IStr (x1,x2,x3,x4,x5,x6,x7)) = {x1,x2,x3,x4,x5,x6,x7} by ThSTC0IS3;
hence ( x1 in InputVertices (STC0IStr (x1,x2,x3,x4,x5,x6,x7)) & x2 in InputVertices (STC0IStr (x1,x2,x3,x4,x5,x6,x7)) & x3 in InputVertices (STC0IStr (x1,x2,x3,x4,x5,x6,x7)) & x4 in InputVertices (STC0IStr (x1,x2,x3,x4,x5,x6,x7)) & x5 in InputVertices (STC0IStr (x1,x2,x3,x4,x5,x6,x7)) & x6 in InputVertices (STC0IStr (x1,x2,x3,x4,x5,x6,x7)) & x7 in InputVertices (STC0IStr (x1,x2,x3,x4,x5,x6,x7)) ) by ENUMSET1:def 5; :: thesis: verum