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
now :: thesis: for x being Element of [:(UnionSt (s,t)),( the Symbols of s \/ the Symbols of t):] holds f . x = g . x
let 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;
hence f = g by FUNCT_2:63; :: thesis: verum