let x1, x2 be object ; :: according to FUNCT_1:def 4 :: thesis: ( not x1 in dom (Special_Function (n1,n2)) or not x2 in dom (Special_Function (n1,n2)) or not (Special_Function (n1,n2)) . x1 = (Special_Function (n1,n2)) . x2 or x1 = x2 )
assume that
A1: x1 in dom (Special_Function (n1,n2)) and
A2: x2 in dom (Special_Function (n1,n2)) ; :: thesis: ( not (Special_Function (n1,n2)) . x1 = (Special_Function (n1,n2)) . x2 or x1 = x2 )
assume A3: (Special_Function (n1,n2)) . x1 = (Special_Function (n1,n2)) . x2 ; :: thesis: x1 = x2
reconsider x1 = x1 as Element of NAT by A1;
reconsider x2 = x2 as Element of NAT by A2;
A4: ( (Special_Function (n1,n2)) . x2 = IFGT (x2,n1,(x2 + n2),x2) & (Special_Function (n1,n2)) . x1 = IFGT (x1,n1,(x1 + n2),x1) ) by Def2;

per cases ( ( x1 <= n1 & x2 <= n1 ) or ( x1 <= n1 & x2 > n1 ) or ( x2 <= n1 & x1 > n1 ) or ( x1 > n1 & x2 > n1 ) ) ;

suppose ( x1 <= n1 & x2 <= n1 ) ; :: thesis: x1 = x2

then ( IFGT (x2,n1,(x2 + n2),x2) = x2 & IFGT (x1,n1,(x1 + n2),x1) = x1 ) by XXREAL_0:def 11;
hence x1 = x2 by A4, A3; :: thesis: verum

end;

suppose A6: ( x1 <= n1 & x2 > n1 ) ; :: thesis: x1 = x2

then IFGT (x2,n1,(x2 + n2),x2) = x2 + n2 by XXREAL_0:def 11;
then A7: (Special_Function (n1,n2)) . x2 = x2 + n2 by Def2;
IFGT (x1,n1,(x1 + n2),x1) = x1 by A6, XXREAL_0:def 11;
then A9: (Special_Function (n1,n2)) . x1 = x1 by Def2;
( x1 <> x2 implies (Special_Function (n1,n2)) . x1 <> (Special_Function (n1,n2)) . x2 )

proof

assume x1 <> x2 ; :: thesis: (Special_Function (n1,n2)) . x1 <> (Special_Function (n1,n2)) . x2
( x1 < x2 & 0 <= n2 ) by A6, XXREAL_0:2;
hence (Special_Function (n1,n2)) . x1 <> (Special_Function (n1,n2)) . x2 by A9, A7, XREAL_1:31; :: thesis: verum

end;

hence x1 = x2 by A3; :: thesis: verum

end;

suppose A10: ( x2 <= n1 & x1 > n1 ) ; :: thesis: x1 = x2

(Special_Function (n1,n2)) . x1 = IFGT (x1,n1,(x1 + n2),x1) by Def2;
then A12: (Special_Function (n1,n2)) . x1 = x1 + n2 by A10, XXREAL_0:def 11;
(Special_Function (n1,n2)) . x2 = IFGT (x2,n1,(x2 + n2),x2) by Def2;
then A14: (Special_Function (n1,n2)) . x2 = x2 by A10, XXREAL_0:def 11;
( x2 <> x1 implies (Special_Function (n1,n2)) . x2 <> (Special_Function (n1,n2)) . x1 )

proof

assume x2 <> x1 ; :: thesis: (Special_Function (n1,n2)) . x2 <> (Special_Function (n1,n2)) . x1
( x2 < x1 & 0 <= n2 ) by A10, XXREAL_0:2;
hence (Special_Function (n1,n2)) . x2 <> (Special_Function (n1,n2)) . x1 by A14, A12, XREAL_1:31; :: thesis: verum

end;

hence x1 = x2 by A3; :: thesis: verum

end;

suppose A15: ( x1 > n1 & x2 > n1 ) ; :: thesis: x1 = x2

( IFGT (x2,n1,(x2 + n2),x2) = x2 + n2 & IFGT (x1,n1,(x1 + n2),x1) = x1 + n2 ) by A15, XXREAL_0:def 11;
then x2 + n2 = x1 + n2 by A4, A3;
hence x1 = x2 ; :: thesis: verum

end;

end;