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;
