let p be Element of HP-WFF ; :: thesis: ( p . 1 = 1 implies p is conditional )
assume A1: p . 1 = 1 ; :: thesis: p is conditional

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 conditional

then consider r, s being Element of HP-WFF such that
A2: p = r '&' s ;
p = <*2*> ^ (r ^ s) by A2, FINSEQ_1:32;
hence p is conditional by A1, FINSEQ_1:41; :: thesis: verum

end;

suppose p is conditional ; :: thesis: p is conditional

hence p is conditional ; :: thesis: verum

end;

suppose p is simple ; :: thesis: p is conditional

then consider n being Element of NAT such that
A3: p = prop n ;
1 + 0 = 1 + (2 + n) by A1, A3;
hence p is conditional ; :: thesis: verum

end;

suppose p = VERUM ; :: thesis: p is conditional

hence p is conditional by A1; :: thesis: verum

end;

end;