let X be set ; :: thesis: for l being Element of MapsC X ex g being Element of FuncsC X ex C1, C2 being Element of CSp X st
( l = [[C1,C2],g] & ( union C2 = {} implies union C1 = {} ) & g is Function of (union C1),(union C2) & ( for x, y being set st {x,y} in C1 holds
{(g . x),(g . y)} in C2 ) )
let l be Element of MapsC X; :: thesis: ex g being Element of FuncsC X ex C1, C2 being Element of CSp X st
( l = [[C1,C2],g] & ( union C2 = {} implies union C1 = {} ) & g is Function of (union C1),(union C2) & ( for x, y being set st {x,y} in C1 holds
{(g . x),(g . y)} in C2 ) )
l in { [[C1,C2],g] where C1, C2 is Element of CSp X, g is Element of FuncsC X : ( ( union C2 = {} implies union C1 = {} ) & g is Function of (union C1),(union C2) & ( for x, y being set st {x,y} in C1 holds
{(g . x),(g . y)} in C2 ) ) } ;
then ex C1, C2 being Element of CSp X ex g being Element of FuncsC X st
( l = [[C1,C2],g] & ( union C2 = {} implies union C1 = {} ) & g is Function of (union C1),(union C2) & ( for x, y being set st {x,y} in C1 holds
{(g . x),(g . y)} in C2 ) ) ;
hence ex g being Element of FuncsC X ex C1, C2 being Element of CSp X st
( l = [[C1,C2],g] & ( union C2 = {} implies union C1 = {} ) & g is Function of (union C1),(union C2) & ( for x, y being set st {x,y} in C1 holds
{(g . x),(g . y)} in C2 ) ) ; :: thesis: verum