set Sm = the Symbols of s \/ the Symbols of t;

set X = [:(UnionSt (s,t)),( the Symbols of s \/ the Symbols of t):];

let f, g be Function of [:(UnionSt (s,t)),( the Symbols of s \/ the Symbols of t):],[:(UnionSt (s,t)),( the Symbols of s \/ the Symbols of t),{(- 1),0,1}:]; :: thesis: ( ( for x being Element of [:(UnionSt (s,t)),( the Symbols of s \/ the Symbols of t):] holds f . x = Uniontran (s,t,x) ) & ( for x being Element of [:(UnionSt (s,t)),( the Symbols of s \/ the Symbols of t):] holds g . x = Uniontran (s,t,x) ) implies f = g )

assume that

A2: for x being Element of [:(UnionSt (s,t)),( the Symbols of s \/ the Symbols of t):] holds f . x = Uniontran (s,t,x) and

A3: for x being Element of [:(UnionSt (s,t)),( the Symbols of s \/ the Symbols of t):] holds g . x = Uniontran (s,t,x) ; :: thesis: f = g

set X = [:(UnionSt (s,t)),( the Symbols of s \/ the Symbols of t):];

let f, g be Function of [:(UnionSt (s,t)),( the Symbols of s \/ the Symbols of t):],[:(UnionSt (s,t)),( the Symbols of s \/ the Symbols of t),{(- 1),0,1}:]; :: thesis: ( ( for x being Element of [:(UnionSt (s,t)),( the Symbols of s \/ the Symbols of t):] holds f . x = Uniontran (s,t,x) ) & ( for x being Element of [:(UnionSt (s,t)),( the Symbols of s \/ the Symbols of t):] holds g . x = Uniontran (s,t,x) ) implies f = g )

assume that

A2: for x being Element of [:(UnionSt (s,t)),( the Symbols of s \/ the Symbols of t):] holds f . x = Uniontran (s,t,x) and

A3: for x being Element of [:(UnionSt (s,t)),( the Symbols of s \/ the Symbols of t):] holds g . x = Uniontran (s,t,x) ; :: thesis: f = g

now :: thesis: for x being Element of [:(UnionSt (s,t)),( the Symbols of s \/ the Symbols of t):] holds f . x = g . x

hence
f = g
by FUNCT_2:63; :: thesis: verumlet x be Element of [:(UnionSt (s,t)),( the Symbols of s \/ the Symbols of t):]; :: thesis: f . x = g . x

thus f . x = Uniontran (s,t,x) by A2

.= g . x by A3 ; :: thesis: verum

end;thus f . x = Uniontran (s,t,x) by A2

.= g . x by A3 ; :: thesis: verum