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

theorem Th36:
  U3(n)Tran.[0,0]=[1,0,1] & U3(n)Tran.[1,1]=[1,0,1] & U3(n)Tran.[1
  ,0]=[2,0,1] & U3(n)Tran.[2,1]=[2,0,1] & U3(n)Tran.[2,0]=[3,0,0]
proof
  set x=[1,1];
  set x1=[0,0];
  set p1=[0,0] .--> [1,0,1], p2=[1,1] .--> [1,0,1], p3=[1,0] .--> [2,0,1], p4=
  [2,1] .--> [2,0,1], f= id([: SegM 3,{0,1} :],{ -1,0,1 },0);
  thus U3(n)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 U3(n)Tran.x=(f +* p1+* p2+* p3+* p4).x by Th2
    .=(f +* p1+* p2+* p3).x by Th2
    .=(f +* p1+* p2).x by Th3
    .=[1,0,1] by FUNCT_7:94;
  set x=[1,0];
  thus U3(n)Tran.x=(f +* p1+* p2+* p3+* p4).x by Th2
    .=(f +* p1+* p2+* p3).x by Th3
    .=[2,0,1] by FUNCT_7:94;
  set x=[2,1];
  thus U3(n)Tran.x=(f +* p1+* p2+* p3+* p4).x by Th3
    .=[2,0,1] by FUNCT_7:94;
  thus thesis by FUNCT_7:94;
end;
