let p be Element of HP-WFF ; :: thesis: ( p . 1 = 2 implies p is conjunctive )
assume A1: p . 1 = 2 ; :: thesis: p is conjunctive
per cases ( p is conjunctive or p is conditional or p is simple or p = VERUM ) by Th9;
suppose p is conjunctive ; :: thesis: p is conjunctive
hence p is conjunctive ; :: thesis: verum
end;
suppose p is conditional ; :: thesis: p is conjunctive
then consider r, s being Element of HP-WFF such that
A2: p = r => s ;
p = <*1*> ^ (r ^ s) by A2, FINSEQ_1:32;
hence p is conjunctive by A1, FINSEQ_1:41; :: thesis: verum
end;
suppose p is simple ; :: thesis: p is conjunctive
then consider n being Element of NAT such that
A3: p = prop n ;
2 + 0 = 2 + (1 + n) by A1, A3, FINSEQ_1:40;
hence p is conjunctive ; :: thesis: verum
end;
suppose p = VERUM ; :: thesis: p is conjunctive
hence p is conjunctive by A1, FINSEQ_1:40; :: thesis: verum
end;
end;