reserve Al for QC-alphabet;
reserve p,q,p1,p2,q1 for Element of CQC-WFF(Al),
  k for Element of NAT,
  f,f1,f2,g for FinSequence of CQC-WFF(Al),
  a,b,b1,b2,c,i,n for Nat;

theorem Th5:
  seq(a,b) \/ {a+b+1} = seq(a,b+1)
proof
  thus seq(a,b) \/ {a+b+1} c= seq(a,b+1)
  proof
    b+0 <= b+1 by XREAL_1:7;
    then
A1: seq(a,b) c= seq(a,b+1) by Th4;
    let x be object;
    assume x in seq(a,b) \/ {a+b+1};
    then x in seq(a,b) or x in {a+b+1} by XBOOLE_0:def 3;
    then x in seq(a,b+1) or x = a+(b+1) by A1,TARSKI:def 1;
    hence thesis by Th3;
  end;
  let x be object such that
A2: x in seq(a,b+1);
  reconsider x as Element of NAT by A2;
  x <= b+1+a by A2,Th1;
  then
A3: x <= a+b or x = a+b+1 by NAT_1:8;
  1+a <= x by A2,Th1;
  then x in seq(a,b) or x in {a+b+1} by A3,TARSKI:def 1;
  hence thesis by XBOOLE_0:def 3;
end;
