let x, y be ExtReal; :: thesis: |.(x * y).| = |.x.| * |.y.|
reconsider x = x, y = y as R_eal by XXREAL_0:def 1;
per cases ( x = 0 or 0 < x or x < 0 ) ;
suppose x = 0 ; :: thesis: |.(x * y).| = |.x.| * |.y.|
then ( |.x.| = 0 & |.(x * y).| = 0 ) by Def1;
hence |.(x * y).| = |.x.| * |.y.| ; :: thesis: verum
end;
suppose A1: 0 < x ; :: thesis: |.(x * y).| = |.x.| * |.y.|
per cases ( y = 0 or 0 < y or y < 0 ) ;
suppose y = 0 ; :: thesis: |.(x * y).| = |.x.| * |.y.|
then ( |.y.| = 0 & |.(x * y).| = 0 ) by Def1;
hence |.(x * y).| = |.x.| * |.y.| ; :: thesis: verum
end;
suppose 0 < y ; :: thesis: |.(x * y).| = |.x.| * |.y.|
then A2: |.y.| = y by Def1;
|.x.| = x by A1, Def1;
hence |.(x * y).| = |.x.| * |.y.| by A2, Def1; :: thesis: verum
end;
suppose A3: y < 0 ; :: thesis: |.(x * y).| = |.x.| * |.y.|
then A4: |.y.| = - y by Def1;
|.x.| = x by A1, Def1;
then |.x.| * |.y.| = - (x * y) by A4, XXREAL_3:92;
hence |.(x * y).| = |.x.| * |.y.| by A1, A3, Def1; :: thesis: verum
end;
end;
end;
suppose A5: x < 0 ; :: thesis: |.(x * y).| = |.x.| * |.y.|
per cases ( y = 0 or 0 < y or y < 0 ) ;
suppose y = 0 ; :: thesis: |.(x * y).| = |.x.| * |.y.|
then ( |.y.| = 0 & |.(x * y).| = 0 ) by Def1;
hence |.(x * y).| = |.x.| * |.y.| ; :: thesis: verum
end;
suppose A6: 0 < y ; :: thesis: |.(x * y).| = |.x.| * |.y.|
then |.y.| = y by Def1;
then A7: |.x.| * |.y.| = (- x) * y by A5, Def1;
|.(x * y).| = - (x * y) by A5, A6, Def1;
hence |.(x * y).| = |.x.| * |.y.| by A7, XXREAL_3:92; :: thesis: verum
end;
suppose y < 0 ; :: thesis: |.(x * y).| = |.x.| * |.y.|
then |.y.| = - y by Def1;
then |.x.| * |.y.| = (- x) * (- y) by A5, Def1;
then |.x.| * |.y.| = - (x * (- y)) by XXREAL_3:92
.= - (- (x * y)) by XXREAL_3:92 ;
hence |.(x * y).| = |.x.| * |.y.| by Def1; :: thesis: verum
end;
end;
end;
end;