let f, g be Function of R^1,(Tunit_circle 2); :: thesis: ( ( for x being Real holds f . x = |[(cos ((2 * PI) * x)),(sin ((2 * PI) * x))]| ) & ( for x being Real holds g . x = |[(cos ((2 * PI) * x)),(sin ((2 * PI) * x))]| ) implies f = g )

assume that

A3: for x being Real holds f . x = |[(cos ((2 * PI) * x)),(sin ((2 * PI) * x))]| and

A4: for x being Real holds g . x = |[(cos ((2 * PI) * x)),(sin ((2 * PI) * x))]| ; :: thesis: f = g

for x being Point of R^1 holds f . x = g . x

assume that

A3: for x being Real holds f . x = |[(cos ((2 * PI) * x)),(sin ((2 * PI) * x))]| and

A4: for x being Real holds g . x = |[(cos ((2 * PI) * x)),(sin ((2 * PI) * x))]| ; :: thesis: f = g

for x being Point of R^1 holds f . x = g . x

proof

hence
f = g
; :: thesis: verum
let x be Point of R^1; :: thesis: f . x = g . x

thus f . x = |[(cos ((2 * PI) * x)),(sin ((2 * PI) * x))]| by A3

.= g . x by A4 ; :: thesis: verum

end;thus f . x = |[(cos ((2 * PI) * x)),(sin ((2 * PI) * x))]| by A3

.= g . x by A4 ; :: thesis: verum