reserve n,i,j,k for Nat;
reserve T for TuringStr,
  s for All-State of T;

theorem Th14:
  Sum_Tran.[0,0]=[0,0,1] & Sum_Tran.[0,1]=[1,0,1] & Sum_Tran.[1,1]
=[1,1,1] & Sum_Tran.[1,0]=[2,1,1] & Sum_Tran.[2,1]=[2,1,1] & Sum_Tran.[2,0]=[3,
0,-1] & Sum_Tran.[3,1]=[4,0,-1] & Sum_Tran.[4,1]=[4,1,-1] & Sum_Tran.[4,0]=[5,0
  ,0]
proof
  set x=[0,1];
  set x1=[0,0];
  set p1=[0,0] .--> [0,0,1], p2=[0,1] .--> [1,0,1], p3=[1,1] .--> [1,1,1], p4=
  [1,0] .--> [2,1,1], p5=[2,1] .--> [2,1,1], p6=[2,0] .--> [3,0,-1], p7=[3,1]
.--> [4,0,-1], p8=[4,1] .--> [4,1,-1], f= id([: SegM 5,{0,1} :],{ -1,0,1 },0);
  thus Sum_Tran.x1=(f +* p1+* p2+* p3+* p4+* p5+* p6+* p7+* p8).x1 by Th2
    .=(f +* p1+* p2+* p3+* p4+* p5+* p6+* p7).x1 by Th2
    .=(f +* p1+* p2+* p3+* p4+* p5+* p6).x1 by Th2
    .=(f +* p1+* p2+* p3+* p4+* p5).x1 by Th2
    .=(f +* p1+* p2+* p3+* p4).x1 by Th2
    .=(f +* p1+* p2+* p3).x1 by Th2
    .=(f +* p1+* p2).x1 by Th2
    .=(f +* p1).x1 by Th3
    .=[0,0,1] by FUNCT_7:94;
  thus Sum_Tran.x=(f +* p1+* p2+* p3+* p4+* p5+* p6+* p7+* p8).x by Th2
    .=(f +* p1+* p2+* p3+* p4+* p5+* p6+* p7).x by Th2
    .=(f +* p1+* p2+* p3+* p4+* p5+* p6).x by Th2
    .=(f +* p1+* p2+* p3+* p4+* p5).x by Th2
    .=(f +* p1+* p2+* p3+* p4).x by Th2
    .=(f +* p1+* p2+* p3).x by Th2
    .=(f +* p1+* p2).x by Th2
    .=[1,0,1] by FUNCT_7:94;
  set x=[1,1];
  thus Sum_Tran.x=(f +* p1+* p2+* p3+* p4+* p5+* p6+* p7+* p8).x by Th2
    .=(f +* p1+* p2+* p3+* p4+* p5+* p6+* p7).x by Th2
    .=(f +* p1+* p2+* p3+* p4+* p5+* p6).x by Th2
    .=(f +* p1+* p2+* p3+* p4+* p5).x by Th2
    .=(f +* p1+* p2+* p3+* p4).x by Th2
    .=(f +* p1+* p2+* p3).x by Th3
    .=[1,1,1] by FUNCT_7:94;
  set x=[1,0];
  thus Sum_Tran.x=(f +* p1+* p2+* p3+* p4+* p5+* p6+* p7+* p8).x by Th2
    .=(f +* p1+* p2+* p3+* p4+* p5+* p6+* p7).x by Th2
    .=(f +* p1+* p2+* p3+* p4+* p5+* p6).x by Th2
    .=(f +* p1+* p2+* p3+* p4+* p5).x by Th2
    .=(f +* p1+* p2+* p3+* p4).x by Th2
    .=[2,1,1] by FUNCT_7:94;
  set x=[2,1];
  thus Sum_Tran.x=(f +* p1+* p2+* p3+* p4+* p5+* p6+* p7+* p8).x by Th2
    .=(f +* p1+* p2+* p3+* p4+* p5+* p6+* p7).x by Th2
    .=(f +* p1+* p2+* p3+* p4+* p5+* p6).x by Th2
    .=(f +* p1+* p2+* p3+* p4+* p5).x by Th3
    .=[2,1,1] by FUNCT_7:94;
  set x=[2,0];
  thus Sum_Tran.x=(f +* p1+* p2+* p3+* p4+* p5+* p6+* p7+* p8).x by Th2
    .=(f +* p1+* p2+* p3+* p4+* p5+* p6+* p7).x by Th2
    .=(f +* p1+* p2+* p3+* p4+* p5+* p6).x by Th2
    .=[3,0,-1] by FUNCT_7:94;
  set x=[3,1];
  thus Sum_Tran.x=(f +* p1+* p2+* p3+* p4+* p5+* p6+* p7+* p8).x by Th2
    .=(f +* p1+* p2+* p3+* p4+* p5+* p6+* p7).x by Th2
    .=[4,0,-1] by FUNCT_7:94;
  set x=[4,1];
  thus Sum_Tran.x=(f +* p1+* p2+* p3+* p4+* p5+* p6+* p7+* p8).x by Th3
    .=[4,1,-1] by FUNCT_7:94;
  thus thesis by FUNCT_7:94;
end;
