J1 ^ J2 is Jordan-block-yielding

proof

let i be Nat; :: according to MATRIXJ2:def 2 :: thesis: ( i in dom (J1 ^ J2) implies ex L being Element of K ex n being Nat st (J1 ^ J2) . i = Jordan_block (L,n) )
assume A1: i in dom (J1 ^ J2) ; :: thesis: ex L being Element of K ex n being Nat st (J1 ^ J2) . i = Jordan_block (L,n)

per cases ( i in dom J1 or ex n being Nat st
( n in dom J2 & i = (len J1) + n ) ) by A1, FINSEQ_1:25;

suppose A2: i in dom J1 ; :: thesis: ex L being Element of K ex n being Nat st (J1 ^ J2) . i = Jordan_block (L,n)

then (J1 ^ J2) . i = J1 . i by FINSEQ_1:def 7;
hence ex L being Element of K ex n being Nat st (J1 ^ J2) . i = Jordan_block (L,n) by A2, Def2; :: thesis: verum

end;

suppose ex n being Nat st
( n in dom J2 & i = (len J1) + n ) ; :: thesis: ex L being Element of K ex n being Nat st (J1 ^ J2) . i = Jordan_block (L,n)

then consider k being Nat such that
A3: k in dom J2 and
A4: i = (len J1) + k ;
(J1 ^ J2) . i = J2 . k by A3, A4, FINSEQ_1:def 7;
hence ex L being Element of K ex n being Nat st (J1 ^ J2) . i = Jordan_block (L,n) by A3, Def2; :: thesis: verum

end;

end;

end;

hence J1 ^ J2 is FinSequence_of_Jordan_block of K ; :: thesis: verum