let f, g be Function of R^1,(Tunit_circle 3); :: thesis: ( ( for x being Real holds f . x = |[(cos ((2 * PI) * x)),(sin ((2 * PI) * x)),0]| ) & ( for x being Real holds g . x = |[(cos ((2 * PI) * x)),(sin ((2 * PI) * x)),0]| ) implies f = g )
assume that
A3: for x being Real holds f . x = |[(cos ((2 * PI) * x)),(sin ((2 * PI) * x)),0]| and
A4: for x being Real holds g . x = |[(cos ((2 * PI) * x)),(sin ((2 * PI) * x)),0]| ; :: thesis: f = g
let x be Point of R^1; :: according to FUNCT_2:def 8 :: thesis: f . x = g . x
thus f . x = |[(cos ((2 * PI) * x)),(sin ((2 * PI) * x)),0]| by A3
.= g . x by A4 ; :: thesis: verum