reserve m,j,p,q,n,l for Element of NAT;
reserve e1,e2 for ExtReal;
reserve i for Nat,
        k,k1,k2,j1 for Element of NAT,
        x,x1,x2,y for set;
reserve p1,p2 for FinSequence;
reserve q,q1,q2,q3,q4 for FinSubsequence,
        p1,p2 for FinSequence;
reserve l1 for Nat,
        j2 for Element of NAT;

theorem Th61:
for p1,p2 being FinSequence, q1,q2 being FinSubsequence st q1 c= p1 & q2 c= p2
  ex ss being FinSubsequence st ss = q1 \/ Shift(q2,len p1) &
  dom Seq ss = Seg (len Seq q1 + len Seq q2)
proof
  let p1,p2 be FinSequence, q1,q2 be FinSubsequence;
  assume that
A1: q1 c= p1 and
A2: q2 c= p2;
    consider ss being FinSubsequence such that
A3: ss = q1 \/ Shift(q2,len p1) by A1,Th60;
A6: rng Sgm dom ss = dom ss by FINSEQ_1:50;
A7: dom Seq ss = dom Sgm dom ss by A6,RELAT_1:27
      .= dom Sgm (dom q1 \/ dom Shift(q2,len p1)) by A3,XTUPLE_0:23
      .= dom ((Sgm dom q1)^(Sgm dom Shift(q2,len p1))) by A1,A2,Lm8
      .= Seg (len Sgm dom q1 + len Sgm dom Shift(q2,len p1))
    by FINSEQ_1:def 7;
A8: len Sgm dom Shift(q2,len p1) = card dom Shift(q2,len p1) by FINSEQ_3:39;
A9: len Sgm dom q1 = card dom q1 by FINSEQ_3:39;
    len Sgm dom Shift(q2,len p1) = card Shift(q2,len p1) by A8,CARD_1:62;
    then
A10: len Sgm dom Shift(q2,len p1) = card q2 by Th41;
A11: len Sgm dom q1 = card q1 by A9,CARD_1:62;
A12: len Sgm dom Shift(q2,len p1) = len Seq q2 by A10,FINSEQ_3:158;
    len Sgm dom q1 = len Seq q1 by A11,FINSEQ_3:158;
    hence thesis by A3,A7,A12;
end;
