let F be Function of T,R^1; :: thesis: ( F = f (#) r implies F is continuous )

assume A4: F = f (#) r ; :: thesis: F is continuous

consider g being Function of T,R^1 such that

A5: for p being Point of T

for s being Real st f . p = s holds

g . p = r * s and

A6: g is continuous by JGRAPH_2:23;

F = g

assume A4: F = f (#) r ; :: thesis: F is continuous

consider g being Function of T,R^1 such that

A5: for p being Point of T

for s being Real st f . p = s holds

g . p = r * s and

A6: g is continuous by JGRAPH_2:23;

F = g

proof

hence
F is continuous
by A6; :: thesis: verum
let x be Point of T; :: according to FUNCT_2:def 8 :: thesis: F . x = g . x

thus F . x = (f . x) * r by A4, VALUED_1:6

.= g . x by A5 ; :: thesis: verum

end;thus F . x = (f . x) * r by A4, VALUED_1:6

.= g . x by A5 ; :: thesis: verum