let A be non empty set ; :: thesis: for o being OperSymbol of (CatSign A) st ( o `1 = 2 or len (o `2 ) = 3 ) holds
ex a, b, c being Element of A st o = compsym a,b,c
let o be OperSymbol of (CatSign A); :: thesis: ( ( o `1 = 2 or len (o `2 ) = 3 ) implies ex a, b, c being Element of A st o = compsym a,b,c )
assume A1: ( o `1 = 2 or len (o `2 ) = 3 ) ; :: thesis: ex a, b, c being Element of A st o = compsym a,b,c
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 ) or ( o `1 in {2} & o `2 in 3 -tuples_on A & o = [(o `1 ),(o `2 )] ) ) by MCART_1:10, MCART_1:23;
then consider a, b, c being set such that
A3: ( a in A & b in A & c in A & o `2 = <*a,b,c*> ) by A1, Th12, FINSEQ_1:def 18, TARSKI:def 1;
reconsider a = a, b = b, c = c as Element of A by A3;
take a ; :: thesis: ex b, c being Element of A st o = compsym a,b,c
take b ; :: thesis: ex c being Element of A st o = compsym a,b,c
take c ; :: thesis: o = compsym a,b,c
thus o = compsym a,b,c by A1, A2, A3, FINSEQ_1:def 18, TARSKI:def 1; :: thesis: verum