let p1, p2 be Point of (TOP-REAL 2); :: thesis: ( p1 `1 <> p2 `1 & p1 `2 <> p2 `2 implies ex p being Point of (TOP-REAL 2) st
( p in LSeg (p1,p2) & p `1 <> p1 `1 & p `1 <> p2 `1 & p `2 <> p1 `2 & p `2 <> p2 `2 ) )
assume that
A1: p1 `1 <> p2 `1 and
A2: p1 `2 <> p2 `2 ; :: thesis: ex p being Point of (TOP-REAL 2) st
( p in LSeg (p1,p2) & p `1 <> p1 `1 & p `1 <> p2 `1 & p `2 <> p1 `2 & p `2 <> p2 `2 )
take p = (1 / 2) * (p1 + p2); :: thesis: ( p in LSeg (p1,p2) & p `1 <> p1 `1 & p `1 <> p2 `1 & p `2 <> p1 `2 & p `2 <> p2 `2 )
A3: p = ((1 - (1 / 2)) * p1) + ((1 / 2) * p2) by EUCLID:32;
hence p in LSeg (p1,p2) ; :: thesis: ( p `1 <> p1 `1 & p `1 <> p2 `1 & p `2 <> p1 `2 & p `2 <> p2 `2 )

hereby :: thesis: ( p `1 <> p2 `1 & p `2 <> p1 `2 & p `2 <> p2 `2 )

assume A4: p `1 = p1 `1 ; :: thesis: contradiction
p `1 = (((1 - (1 / 2)) * p1) `1) + (((1 / 2) * p2) `1) by A3, TOPREAL3:2
.= ((1 - (1 / 2)) * (p1 `1)) + (((1 / 2) * p2) `1) by TOPREAL3:4
.= ((1 - (1 / 2)) * (p `1)) + ((1 / 2) * (p2 `1)) by A4, TOPREAL3:4 ;
hence contradiction by A1, A4; :: thesis: verum

end;

hereby :: thesis: ( p `2 <> p1 `2 & p `2 <> p2 `2 )

assume A5: p `1 = p2 `1 ; :: thesis: contradiction
p `1 = (((1 - (1 / 2)) * p1) `1) + (((1 / 2) * p2) `1) by A3, TOPREAL3:2
.= ((1 - (1 / 2)) * (p1 `1)) + (((1 / 2) * p2) `1) by TOPREAL3:4
.= ((1 - (1 / 2)) * (p1 `1)) + ((1 / 2) * (p `1)) by A5, TOPREAL3:4 ;
hence contradiction by A1, A5; :: thesis: verum

end;

hereby :: thesis: p `2 <> p2 `2

assume A6: p `2 = p1 `2 ; :: thesis: contradiction
p `2 = (((1 - (1 / 2)) * p1) `2) + (((1 / 2) * p2) `2) by A3, TOPREAL3:2
.= ((1 - (1 / 2)) * (p1 `2)) + (((1 / 2) * p2) `2) by TOPREAL3:4
.= ((1 - (1 / 2)) * (p `2)) + ((1 / 2) * (p2 `2)) by A6, TOPREAL3:4 ;
hence contradiction by A2, A6; :: thesis: verum

end;

hereby :: thesis: verum

assume A7: p `2 = p2 `2 ; :: thesis: contradiction
p `2 = (((1 - (1 / 2)) * p1) `2) + (((1 / 2) * p2) `2) by A3, TOPREAL3:2
.= ((1 - (1 / 2)) * (p1 `2)) + (((1 / 2) * p2) `2) by TOPREAL3:4
.= ((1 - (1 / 2)) * (p1 `2)) + ((1 / 2) * (p `2)) by A7, TOPREAL3:4 ;
hence contradiction by A2, A7; :: thesis: verum

end;