let r be Complex; :: thesis: for seq being Complex_Sequence st r <> 0c & seq is non-zero holds

r (#) seq is non-zero

let seq be Complex_Sequence; :: thesis: ( r <> 0c & seq is non-zero implies r (#) seq is non-zero )

assume that

A1: r <> 0c and

A2: seq is non-zero and

A3: not r (#) seq is non-zero ; :: thesis: contradiction

consider n1 being Element of NAT such that

A4: (r (#) seq) . n1 = 0c by A3, Th4;

A5: r * (seq . n1) = 0c by A4, VALUED_1:6;

seq . n1 <> 0c by A2, Th4;

hence contradiction by A1, A5; :: thesis: verum

r (#) seq is non-zero

let seq be Complex_Sequence; :: thesis: ( r <> 0c & seq is non-zero implies r (#) seq is non-zero )

assume that

A1: r <> 0c and

A2: seq is non-zero and

A3: not r (#) seq is non-zero ; :: thesis: contradiction

consider n1 being Element of NAT such that

A4: (r (#) seq) . n1 = 0c by A3, Th4;

A5: r * (seq . n1) = 0c by A4, VALUED_1:6;

seq . n1 <> 0c by A2, Th4;

hence contradiction by A1, A5; :: thesis: verum