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

theorem Th33:
  Zero_Tran.[0,0]=[1,0,1] & Zero_Tran.[1,1]=[2,1,1] & Zero_Tran.[2
  ,0]=[3,0,-1] & Zero_Tran.[2,1]=[3,0,-1] & Zero_Tran.[3,1]=[4,1,-1]
proof
  set x=[1,1];
  set x1=[0,0];
  set p1=[0,0] .--> [1,0,1], p2=[1,1] .--> [2,1,1], p3=[2,0] .--> [3,0,-1], p4
  =[2,1] .--> [3,0,-1], f= id([: SegM 4,{0,1} :],{ -1,0,1 },1);
  thus Zero_Tran.x1=(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
    .=[1,0,1] by FUNCT_7:94;
  thus Zero_Tran.x=(f +* p1+* p2+* p3+* p4).x by Th2
    .=(f +* p1+* p2+* p3).x by Th2
    .=(f +* p1+* p2).x by Th3
    .=[2,1,1] by FUNCT_7:94;
  set x=[2,0];
  thus Zero_Tran.x=(f +* p1+* p2+* p3+* p4).x by Th3
    .=(f +* p1+* p2+* p3).x by Th3
    .=[3,0,-1] by FUNCT_7:94;
  set x=[2,1];
  thus Zero_Tran.x=(f +* p1+* p2+* p3+* p4).x by Th2
    .=[3,0,-1] by FUNCT_7:94;
  thus thesis by FUNCT_7:94;
end;
