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 Def3;

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_1:21;

then consider a, b, c being object such that

A3: ( a in A & b in A & c in A ) and

A4: o `2 = <*a,b,c*> by A1, CARD_1:def 7, FINSEQ_2:139, 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, A4, CARD_1:def 7, TARSKI:def 1; :: thesis: verum

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 Def3;

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_1:21;

then consider a, b, c being object such that

A3: ( a in A & b in A & c in A ) and

A4: o `2 = <*a,b,c*> by A1, CARD_1:def 7, FINSEQ_2:139, 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, A4, CARD_1:def 7, TARSKI:def 1; :: thesis: verum