let A be non empty set ; :: thesis: for o being OperSymbol of (CatSign A) st ( o `1 = 1 or len (o `2 ) = 1 ) holds
ex a being Element of A st o = idsym a
let o be OperSymbol of (CatSign A); :: thesis: ( ( o `1 = 1 or len (o `2 ) = 1 ) implies ex a being Element of A st o = idsym a )
assume A1: ( o `1 = 1 or len (o `2 ) = 1 ) ; :: thesis: ex a being Element of A st o = idsym a
the carrier' of (CatSign A) = [:{1},(1 -tuples_on A):] \/ [:{2},(3 -tuples_on A):] by Def5;
then ( o in [:{1},(1 -tuples_on A):] or o in [:{2},(3 -tuples_on A):] ) by XBOOLE_0:def 3;
then A2: ( ( o `1 in {1} & o `2 in 1 -tuples_on A & o = [(o `1 ),(o `2 )] ) or ( o `1 in {2} & o `2 in 3 -tuples_on A ) ) by MCART_1:10, MCART_1:23;
then consider a being set such that
A3: ( a in A & o `2 = <*a*> ) by A1, Th8, FINSEQ_1:def 18, TARSKI:def 1;
reconsider a = a as Element of A by A3;
take a ; :: thesis: o = idsym a
thus o = idsym a by A1, A2, A3, FINSEQ_1:def 18, TARSKI:def 1; :: thesis: verum