reserve i,j,k,m,n for Nat,
  D for non empty set,
  p for Element of D,
  f for FinSequence of D;
reserve D for non empty set,
  p for Element of D,
  f for FinSequence of D;
reserve f for circular FinSequence of D;
reserve f,g for FinSequence of TOP-REAL 2;
reserve p for Point of TOP-REAL 2,
  f for FinSequence of TOP-REAL 2;
reserve f for circular FinSequence of TOP-REAL 2;

theorem Th36:
  for f being non constant standard special_circular_sequence
  holds LeftComp Rotate(f,p) = LeftComp f
proof
  let f be non constant standard special_circular_sequence;
A1: p in rng f implies p..f >= 1 by FINSEQ_4:21;
  LeftComp Rotate(f,p) is_a_component_of (L~Rotate(f,p))` by GOBOARD9:def 1;
  then
A2: LeftComp Rotate(f,p) is_a_component_of (L~f)` by Th33;
  per cases by A1,XXREAL_0:1;
  suppose
    not p in rng f;
    hence thesis by FINSEQ_6:def 2;
  end;
  suppose
    p in rng f & p..f = 1;
    then 1 in dom f & f.1 = p by FINSEQ_4:19,FINSEQ_5:6;
    then f/.1 = p by PARTFUN1:def 6;
    hence thesis by FINSEQ_6:89;
  end;
  suppose that
A3: p in rng f and
A4: 1 < p..f;
A5: p..f <= len f by A3,FINSEQ_4:21;
    then
A6: 1 + p..f <= 1 + len f by XREAL_1:6;
    1 + 1 <= p..f by A4,NAT_1:13;
    then 1 + 1 + len f <= len f + p..f by XREAL_1:6;
    then len f <= len f + 1 & 1 + len f + 1 -' p..f <= len f by NAT_1:11
,NAT_D:53;
    then 1 + len f -' p..f + 1 <= len f by A5,NAT_D:38,XXREAL_0:2;
    then
A7: 1 + len f -' p..f + 1 <= len Rotate(f,p) by Th14;
    left_cell(f,1) = left_cell(Rotate(f,p),1 + len f -' p..f) by A3,A4,Th35;
    then Int left_cell(f,1) c= LeftComp Rotate(f,p) by A6,A7,GOBOARD9:21
,NAT_D:55;
    hence thesis by A2,GOBOARD9:def 1;
  end;
end;
