let a, b, c, d, n be Element of NAT ; :: thesis: (GenFib (a,b,n)) + (GenFib (c,d,n)) = GenFib ((a + c),(b + d),n)
defpred S₁[ Nat] means (GenFib (a,b,$1)) + (GenFib (c,d,$1)) = GenFib ((a + c),(b + d),$1);
A1: for k being Nat st S₁[k] & S₁[k + 1] holds
S₁[k + 2]

proof

let k be Nat; :: thesis: ( S₁[k] & S₁[k + 1] implies S₁[k + 2] )
assume that
A2: S₁[k] and
A3: S₁[k + 1] ; :: thesis: S₁[k + 2]
(GenFib (a,b,(k + 2))) + (GenFib (c,d,(k + 2))) = ((GenFib (a,b,k)) + (GenFib (a,b,(k + 1)))) + (GenFib (c,d,(k + 2))) by Th34
.= ((GenFib (a,b,k)) + (GenFib (a,b,(k + 1)))) + ((GenFib (c,d,k)) + (GenFib (c,d,(k + 1)))) by Th34
.= (GenFib ((a + c),(b + d),k)) + ((GenFib (a,b,(k + 1))) + (GenFib (c,d,(k + 1)))) by A2
.= GenFib ((a + c),(b + d),(k + 2)) by A3, Th34 ;
hence S₁[k + 2] ; :: thesis: verum

end;

(GenFib (a,b,1)) + (GenFib (c,d,1)) = b + (GenFib (c,d,1)) by Th32
.= b + d by Th32
.= GenFib ((a + c),(b + d),1) by Th32 ;
then A4: S₁[1] ;
(GenFib (a,b,0)) + (GenFib (c,d,0)) = a + (GenFib (c,d,0)) by Th32
.= a + c by Th32
.= GenFib ((a + c),(b + d),0) by Th32 ;
then A5: S₁[ 0 ] ;
for k being Nat holds S₁[k] from FIB_NUM:sch 1(A5, A4, A1);
hence (GenFib (a,b,n)) + (GenFib (c,d,n)) = GenFib ((a + c),(b + d),n) ; :: thesis: verum