reserve A,B,C,Y,x,y,z for set, U, D for non empty set,
X for non empty Subset of D, d,d1,d2 for Element of D;
reserve P,Q,R for Relation, g for Function, p,q for FinSequence;
reserve f for BinOp of D, i,m,n for Nat;
reserve X for set, f for Function;
reserve U1,U2 for non empty set;
reserve f for BinOp of D;
reserve a,a1,a2,b,b1,b2,A,B,C,X,Y,Z,x,x1,x2,y,y1,y2,z for set,
U,U1,U2,U3 for non empty set, u,u1,u2 for Element of U,
P,Q,R for Relation, f,f1,f2,g,g1,g2 for Function,
k,m,n for Nat, kk,mm,nn for Element of NAT, m1, n1 for non zero Nat,
p, p1, p2 for FinSequence, q, q1, q2 for U-valued FinSequence;

theorem for p1,p2,q1,q2 being FinSequence st p1 is m-element & ::#Th41
q1 is m-element & (p1^p2=q1^q2 or p2^p1=q2^q1) holds (p1=q1 & p2=q2)
proof ::#weakening FINSEQ_1:33
let p1,p2,q1,q2 be FinSequence;
set m1=len p1, m2=len p2, n1=len q1, n2=len q2; assume p1 is m-element
& q1 is m-element; then reconsider p11=p1, q11=q1 as m-element
FinSequence; reconsider p22=p2 null p2 as m2-element FinSequence;
reconsider q22=q2 null q2 as n2-element FinSequence;
set PA=p11^p22, PB=p22^p11, QA=q11^q22, QB=q22^q11;
A1: len PA=m+m2 & len PB=m2+m & len QA=m+n2 & len QB=n2+m by CARD_1:def 7;
assume
A2: p1^p2=q1^q2 or p2^p1=q2^q1; then
A3: PA=QA or PB=QB;
reconsider q22 as m2-element FinSequence by A2, A1;
p22 is m2-element & q22 is m2-element;
hence thesis by A3, Lm49;
end;
