let f, g be UnOp of the carrier of F; :: thesis: ( ( for x being Element of F holds f . x = - x ) & ( for x being Element of F holds g . x = - x ) implies f = g )

assume that

A1: for x being Element of F holds f . x = - x and

A2: for x being Element of F holds g . x = - x ; :: thesis: f = g

assume that

A1: for x being Element of F holds f . x = - x and

A2: for x being Element of F holds g . x = - x ; :: thesis: f = g

now :: thesis: for x being object st x in the carrier of F holds

f . x = g . x

hence
f = g
by FUNCT_2:12; :: thesis: verumf . x = g . x

let x be object ; :: thesis: ( x in the carrier of F implies f . x = g . x )

assume x in the carrier of F ; :: thesis: f . x = g . x

then reconsider y = x as Element of F ;

thus f . x = - y by A1

.= g . x by A2 ; :: thesis: verum

end;assume x in the carrier of F ; :: thesis: f . x = g . x

then reconsider y = x as Element of F ;

thus f . x = - y by A1

.= g . x by A2 ; :: thesis: verum