let r be Real; :: thesis: for seq being Real_Sequence st r <> 0 & seq is non-zero holds
r (#) seq is non-zero
let seq be Real_Sequence; :: thesis: ( r <> 0 & seq is non-zero implies r (#) seq is non-zero )
assume that
A1: r <> 0 and
A2: seq is non-zero and
A3: not r (#) seq is non-zero ; :: thesis: contradiction
consider n1 being Nat such that
A4: (r (#) seq) . n1 = 0 by A3, Th5;
A5: seq . n1 <> 0 by A2, Th5;
r * (seq . n1) = 0 by A4, Th9;
hence contradiction by A1, A5, XCMPLX_1:6; :: thesis: verum