reserve a,x,y for object, A,B for set,
  l,m,n for Nat;

theorem Th22:
  for p,q being FinSequence st len p = len q+1
  for i being Nat holds i in dom q iff i in dom p & i+1 in dom p
proof
  let p,q be FinSequence;
  assume
A1: len p = len q+1;
  let i be Nat;
  hereby
    assume
A2: i in dom q;
    then
A3: i >= 1 by FINSEQ_3:25;
A4: i <= len q by A2,FINSEQ_3:25;
    len q <= len p by A1,NAT_1:11;
    then
A5: i+1 >= 1 & i <= len p by A4,NAT_1:11,XXREAL_0:2;
    i+1 <= len p by A1,A4,XREAL_1:6;
    hence i in dom p & i+1 in dom p by A3,A5,FINSEQ_3:25;
  end;
  assume that
A6: i in dom p and
A7: i+1 in dom p;
  i+1 <= len p by A7,FINSEQ_3:25;
  then
A8: i <= len q by A1,XREAL_1:6;
  i >= 1 by A6,FINSEQ_3:25;
  hence thesis by A8,FINSEQ_3:25;
end;
