deffunc H₁( Element of [:F₁(),F₂(),F₃():], Element of [:F₁(),F₂(),F₃():]) -> Element of F₄() = F₅(($1 `1_3),($2 `1_3),($1 `2_3),($2 `2_3),($1 `3_3),($2 `3_3));
consider f being Function of [:[:F₁(),F₂(),F₃():],[:F₁(),F₂(),F₃():]:],F₄() such that
A1: for x, y being Element of [:F₁(),F₂(),F₃():] holds f . (x,y) = H₁(x,y) from BINOP_1:sch 4();
take f ; :: thesis: for x1, y1 being Element of F₁()
for x2, y2 being Element of F₂()
for x3, y3 being Element of F₃()
for x, y being Element of [:F₁(),F₂(),F₃():] st x = [x1,x2,x3] & y = [y1,y2,y3] holds
f . (x,y) = F₅(x1,y1,x2,y2,x3,y3)
let x1, y1 be Element of F₁(); :: thesis: for x2, y2 being Element of F₂()
for x3, y3 being Element of F₃()
for x, y being Element of [:F₁(),F₂(),F₃():] st x = [x1,x2,x3] & y = [y1,y2,y3] holds
f . (x,y) = F₅(x1,y1,x2,y2,x3,y3)
let x2, y2 be Element of F₂(); :: thesis: for x3, y3 being Element of F₃()
for x, y being Element of [:F₁(),F₂(),F₃():] st x = [x1,x2,x3] & y = [y1,y2,y3] holds
f . (x,y) = F₅(x1,y1,x2,y2,x3,y3)
let x3, y3 be Element of F₃(); :: thesis: for x, y being Element of [:F₁(),F₂(),F₃():] st x = [x1,x2,x3] & y = [y1,y2,y3] holds
f . (x,y) = F₅(x1,y1,x2,y2,x3,y3)
let x, y be Element of [:F₁(),F₂(),F₃():]; :: thesis: ( x = [x1,x2,x3] & y = [y1,y2,y3] implies f . (x,y) = F₅(x1,y1,x2,y2,x3,y3) )
assume that
A2: x = [x1,x2,x3] and
A3: y = [y1,y2,y3] ; :: thesis: f . (x,y) = F₅(x1,y1,x2,y2,x3,y3)
A5: ( y1 = y `1_3 & y2 = y `2_3 ) by A3;
A7: x3 = x `3_3 by A2;
A8: y3 = y `3_3 by A3;
( x1 = x `1_3 & x2 = x `2_3 ) by A2;
hence f . (x,y) = F₅(x1,y1,x2,y2,x3,y3) by A1, A5, A7, A8; :: thesis: verum