reserve P for Subset of TOP-REAL 2,
  f,f1,f2,g for FinSequence of TOP-REAL 2,
  p,p1,p2,q,q1,q2 for Point of TOP-REAL 2,
  r1,r2,r19,r29 for Real,
  i,j,k,n for Nat;

theorem Th40:
  f is special implies Rev f is special
proof
  assume
A1: f is special;
A2: len Rev f = len f by FINSEQ_5:def 3;
  let i be Nat such that
A3: 1 <= i and
A4: i+1 <= len(Rev f);
  i <= i+1 by NAT_1:11;
  then reconsider j = len f - i as Element of NAT by A4,A2,INT_1:5,XXREAL_0:2;
  j <= len f - 1 by A3,XREAL_1:10;
  then
A5: j+1 <= len f - 1 + 1 by XREAL_1:6;
A6: 1+i+j = len f + 1;
A7: i+1-i <= j by A4,A2,XREAL_1:9;
  then j in dom f by A5,SEQ_4:134;
  then
A8: (Rev f)/.(i+1) = f/.j by A6,FINSEQ_5:66;
A9: i+(j+1) = len f + 1;
  j+1 in dom f by A5,A7,SEQ_4:134;
  then (Rev f)/.i = f/.(j+1) by A9,FINSEQ_5:66;
  hence thesis by A1,A5,A7,A8;
end;
